Reference Systems forSurveying and MappingLecture notes

Hans van der Marel

DelftUniversityofTechnology

ii

The front cover shows the NAP (Amsterdam Ordnance Datum) ”datum point” at the Stopera,Amsterdam (picture M.M.Minderhoud, Wikipedia/Michiel1972).

H. van der Marel

Lecture notes on Reference Systems for Surveying and Mapping:CTB3310 Surveying and MappingCTB3425 Monitoring and Stability of Dikes and EmbankmentsCIE4606 Geodesy and Remote SensingCIE4614 Land Surveying and Civil Infrastructure

February 2020

Publisher:Faculty of Civil Engineering and GeosciencesDelft University of TechnologyP.O. Box 5048Stevinweg 12628 CN DelftThe Netherlands

Copyright ©2014­2020 by H. van der MarelThe content in these lecture notes, except for material credited to third parties, is licensedunder a Creative Commons Attributions­NonCommercial­SharedAlike 4.0 International License(CC BY­NC­SA). Third party material is shared under its own license and attribution.

The text has been type set using the MikTex 2.9 implementation of LATEX. Graphs and diagramswere produced, if not mentioned otherwise, with Matlab and Inkscape.

Preface

This reader on reference systems for surveying and mapping has been initially compiled forthe course Surveying and Mapping (CTB3310) in the 3rd year of the BSc­program for CivilEngineering. The reader is aimed at students at the end of their BSc program or at the startof their MSc program, and is used in several courses at Delft University of Technology.

With the advent of the Global Positioning System (GPS) technology in mobile (smart)phones and other navigational devices almost anyone, anywhere on Earth, and at any time,can determine a three–dimensional position accurate to a few meters. With some modest in­vestments, basically using the same GPS equipment, the Internet, and correction signals froma network of reference GPS receivers, determined individuals and professional users alike canachieve with relative ease three–dimensional positions with centimeter accuracy. A feat thatuntil recently was achievable only to a small community of land surveyors and geodesists.

Our increased ability to collect accurate positional data, but also advances in GeographicInformation Systems (GIS) and adoption of open–data policies for sharing many geographicdatasets, has resulted in huge amounts of georeferenced data available to users.

However, sharing positional information is not always easy: “How come my position mea­surement does not match yours?”, “You say you have centimeter accuracy, I know I have, andyet we have a hunderd meter difference (you fool)?”. These are just a few frustated outcriesyou can hear from users (including Civil Engineering students). The reason is simple: usersmay have opted for different coordinate reference systems (CRS). Positions are relative, givenwith respect to a specific reference system. There are significant differences between variousreference systems that are used, sometimes for historical reasons, sometimes because usersselected different options (e.g. map projection) for good reasons. The solution is straight­forward, but not simple: knowing the name and identifier for the reference system is key. Ifyou have positional data in the same reference system you are lucky, if not, you have to usecoordinate transformations to convert them into the same reference system.

This reader will provide you with the background information and the terminology that iscommonly used. For the actual transformations you can use (freely) available software.

This reader keeps a quite informal and introductive style. The subjects are not treatedwith full mathematical rigor. This reader has been written with an educational goal in mind.The idea is to give students in Civil Engineering and Geosciences some insight in the referencesystems used for surveying and mapping in and outside the Netherlands. Nevertheless, webelieve this reader will also be usefull for other students and professionals in any of the geo­sciences. Sections marked by a [*] contain advanced material that students Civil Engineeringmay skip for the exam.

Colleague Christian Tiberius is acknowledged for his contributions to Chapter 7 and Ap­pendix A, and for providing many of the exercises. Also I would like to thank Christian Tiberiusand Ramon Hanssen for the initial proofreading. Jochem Lesparre and Christian Tiberius arecredited for many of the improvements to the current release, and Cornelis Slobbe for thematerial on the new Dutch quasi–geoid and LAT chart datum. The author does welcome no­tifications of corrections and suggestions for improvement, as well as feedback in general.

Hans van der MarelDelft, February 2020

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Contents

1 Introduction 1

2 2D Cartesian coordinate systems 32.1 2D Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 2D coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Shape preserving transformations . . . . . . . . . . . . . . . . . . . . . 42.2.2 Affine and polynomial transformations . . . . . . . . . . . . . . . . . . 5

2.3 Realization of 2D coordinate systems . . . . . . . . . . . . . . . . . . . . . . . 52.4 Worked out examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 2D coordinate system definition . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Algebraic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 3D Cartesian coordinate systems 113.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 3D Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 3D similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 Overview 3D coordinate transformations . . . . . . . . . . . . . . . . 133.3.2 7­parameter similarity transformation . . . . . . . . . . . . . . . . . . 143.3.3 7–parameter Helmert (small angle) transformation . . . . . . . . . . 163.3.4 10–parameter Molodensky–Badekas transformation [*] . . . . . . . 16

3.4 Realization of 3D coordinate systems . . . . . . . . . . . . . . . . . . . . . . . 173.5 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Spherical and ellipsoidal coordinate systems 194.1 Spherical coordinates (geocentric coordinates) . . . . . . . . . . . . . . . . . 194.2 Geographic coordinates (ellipsoidal coordinates) . . . . . . . . . . . . . . . . 204.3 Astronomical latitude and longitude [*] . . . . . . . . . . . . . . . . . . . . . . 234.4 Topocentric coordinates, azimuth and zenith angle [*] . . . . . . . . . . . . 244.5 Practical aspects of using latitude and longitude. . . . . . . . . . . . . . . . 274.6 Spherical and ellipsoidal computations [*] . . . . . . . . . . . . . . . . . . . . 294.7 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Map projections 355.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Map projection types and properties . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Practical aspects of map projections . . . . . . . . . . . . . . . . . . . . . . . . 385.4 Cylindrical Map projection examples . . . . . . . . . . . . . . . . . . . . . . . 38

5.4.1 Central cylindrical projection . . . . . . . . . . . . . . . . . . . . . . . . 395.4.2 Mercator projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4.3 Plate carrée and equirectangular projections . . . . . . . . . . . . . . 415.4.4 Web Mercator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4.5 Universal Transverse Mercator (UTM). . . . . . . . . . . . . . . . . . . 42

5.5 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

v

vi Contents

6 Datum transformations and coordinate conversions 456.1 Geodetic datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Coordinate operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3 A brief history of geodetic datums . . . . . . . . . . . . . . . . . . . . . . . . . 486.4 EPSG dataset and coordinate conversion software. . . . . . . . . . . . . . . 486.5 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Gravity and gravity potential 517.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 Earth’s gravity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.3 Gravity Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.4 Geoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.5 Gravimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.7 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Vertical reference systems 578.1 Ellipsoidal heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.2 Orthometric and normal heights . . . . . . . . . . . . . . . . . . . . . . . . . . 588.3 Height datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.4 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9 International reference systems and frames 619.1 World Geodetic System 1984 (WGS84) . . . . . . . . . . . . . . . . . . . . . . 619.2 International Terrestrial Reference System and Frames . . . . . . . . . . . 629.3 European Terrestrial Reference System 1989 (ETRS89) . . . . . . . . . . . 659.4 Problems and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

10Dutch national reference systems 6910.1Dutch Triangulation System (RD). . . . . . . . . . . . . . . . . . . . . . . . . . 69

10.1.1RD1918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7010.1.2RD2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7210.1.3RDNAPTRANS™2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.2Amsterdam Ordnance Datum ­ Normaal Amsterdams Peil (NAP) . . . . . 7510.2.1Precise first order levellings . . . . . . . . . . . . . . . . . . . . . . . . . 7510.2.2NAP Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

10.3Geoid models – NLGEO2004 and NLGEO2018 . . . . . . . . . . . . . . . . . 7810.4Lowest Astronomical Tide (LAT) model – NLLAT2018 . . . . . . . . . . . . . 79

11Further reading 81

A Quantity, dimension, unit 83

1Introduction

Surveying and mapping deal with the description of the shape of the Earth, spatial relationshipsbetween objects near the Earth’s surface, and data associated to these. Mapping is concernedwith the (scaled) portrayal of geographic features and visualization of data in a geographicframework. Mapping is more than the creation of paper maps: contemporary maps are mostlydigital, allowmultiple visualizations and analysis of the data in Geographic Information Systems(GIS). Surveying is concerned with accurately determining the terrestrial or three­dimensionalposition of points, and the distances and angles between them. Points describe the objectsand shapes that appear on maps and in Geographic Information Systems. These points areusually on the surface of the Earth. They are used to connect topographic features, boundariesfor ownership, locations of buildings, location of subsurface features (pipe­lines), and they areoften used to monitor changes (deformation, subsidence). Points may only exist on paper, orin a CAD system, and represent points to be staked out for construction work.

To describe the position of points a mathematical framework is needed. This mathematicalframework consist of a coordinate reference system (CRS). A coordinate system uses one ormore numbers, or coordinates, to uniquely determine the position of a point in a 2D or 3DEuclidean space. In this reader several of these mathematical frameworks are described andhow they are used in surveying and mapping.

In Chapters 2 and 3 two and three dimensional Cartesian coordinate systems are intro­duced. Although straightforward, 3D Cartesian coordinates are not very convenient for de­scribing positions on the surface of the Earth. It is actually more convenient to use curvilinearcoordinates, or, to project the curved surface of the Earth on a flat plane. Curvilinear coordi­nates, known as geographic coordinates, or latitude and longitude, are discussed in Chapter 4.The map projections, which result in easy to use 2D Cartesian coordinates, are covered inChapter 5.

Coordinate conversions and geodetic datum transformations are discussed in Chapter 6.This topic can be somewhat bewildering for the inexperienced user because there are so manydifferent coordinate types and geodetic datums in use, but fortunately any transformationcan be decomposed into a few elementary coordinate conversions and a geodetic datumtransformation.

The height always plays a special role in coordinate reference systems. Height is alsoclosely associated with the flow of water and gravity. The height coordinate systems arediscussed in Chapter 8, while in Chapter 7 basic background information on the Earth’s gravityfield is given.

Finally in Chapter 9 and 10 several important and commonly used reference systemsare described. They include the well know World Geodetic System (WGS84) used by GPS,

1

2 1. Introduction

the International Terrestrial Reference System (ITRS), and the European Terrestrial ReferenceSystem (ETRS89) in Chapter 9, and the Dutch triangulation system (”Rijksdriehoeksstelsel”RD) and the Dutch height system, the Amsterdam Ordnance Datum ( ”Normaal AmsterdamsPeil” NAP), and Lowest Astronomical Tide (LAT) chart datum, in Chapter 10.

This reader was written close, very close, to 52.0000∘ North latitude. The 52­degrees Northlatitude happens to run across the campus of Delft University of Technology, just a few metersNorth of the Civil Engineering and Geosciences faculty building. In 2018, the 52­degrees Northline was visualized at the campus with a blue–white line, see Figure 1.1. Check out https://www.delta.tudelft.nl/article/why­blue­line­running­across­campus for afull story how the line was realized and how it has moved over the campus.

Figure 1.1: The blue–white line on the campus of Delft University of Technology visualizes latitude 𝜑 =52.0000000∘ North. The 16­millimetre­wide black line, in the middle of the white band, indicates the ‘exact’position of the 52∘ latitude, in the International Terrestrial Reference System (ITRS), on 1 January 2018. Timematters because the 52∘ parallel shifts due to plate tectonics. The width of the black line represents the shift peryear. To illustrate the time effect further, six grey lines (not visible in picture) have been painted parallel to theblue line. These grey lines indicate significant events related to Delft events. The first line is 2.79 metres to theNorth, and marks the foundation of TU Delft in 1842. (Picture courtesy of Conny van Uffelen)

22D Cartesian coordinate systems

To describe the position of points on a plane surface, be it a plot of land, a piece of paper,or a computer screen, a two–dimensional (2D) coordinate system need to be defined. Oneof the best known 2D coordinate reference systems is the 2D Cartesian coordinate systemwhich uses rectangular coordinates. 2D Cartesian coordinates can also be the result of a mapprojection. Map projections are discussed in Chapter 5.

2.1. 2D Cartesian coordinatesThe position of a point 𝑃𝑖 on a plane surface can be described by two coordinates, 𝑥𝑖 and 𝑦𝑖,in a two dimensional (2D) Cartesian coordinate system, as illustrated in Figure 2.1a. The axesin the 2D Cartesian coordinate system, named after the 17th century mathematician RenéDescartes, are perpendicular (orthogonal), have the same scale, and meet in what is calledthe origin. The Cartesian coordinate system is right­handed, meaning, with the positive x­axis pointing right, the positive y­axis is pointing up1. Therefore, fixing or choosing one axis,determines the other axis. The coordinates (𝑥𝑖, 𝑦𝑖) are defined as the distance from the originto the perpendicular projection of the point 𝑃𝑖 onto the respective axes. The point 𝑃𝑖 can alsobe represented by a position vector 𝐫𝑖 from the origin to the point 𝑃𝑖,

𝐫𝑖 = 𝑥𝑖𝐞𝑥 + 𝑦𝑖𝐞𝑦 , (2.1)

with 𝐞𝑥 and 𝐞𝑦 the unit vectors defining the axis of the Cartesian system (𝐞𝑥 ⊥ 𝐞𝑦).For surveying and mapping the distance 𝑑12 and azimuth 𝛼12 between two points 𝑃1 and

𝑃2 are defined as,𝑑12 = ‖𝐫2 − 𝐫1‖ = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

𝛼12 = arctan𝑥2 − 𝑥1𝑦2 − 𝑦1

(2.2)

The azimuth 𝛼 is given in angular units (degrees, radians, gon) while the distance 𝑑 is ex­pressed in length units (meters), see also Appendix A. For practical computations the arctanin Eq. (2.2) should be replaced with the atan2 (𝑥2 − 𝑥1, 𝑦2 − 𝑦1) function in order to obtainthe right quadrant for the azimuth 𝛼12 2. The angle ∠𝑃2𝑃1𝑃3 between points 𝑃2, 𝑃1 and 𝑃3 is,

𝜑213 = ∠𝑃2𝑃1𝑃3 = 𝛼13 − 𝛼12 = arccos< 𝐫2 − 𝐫1, 𝐫3 − 𝐫1 >‖𝐫2 − 𝐫1‖‖𝐫3 − 𝐫1‖

(2.3)

1An easy way to remember this is the right hand rule: place your right hand on the plane with the thumb pointingup (in the direction of a ”z­axis”), the fingers now point from the x­axis to the y­axis.2The atan2 function is the four quadrant version of the arctangent function with 2 input values, with −𝜋 ≤atan2 (𝑑𝑥, 𝑑𝑦) ≤ 𝜋, compared to −𝜋/2 ≤ arctan 𝑑𝑥

𝑑𝑦 ≤ 𝜋/2

3

4 2. 2D Cartesian coordinate systems

x

α12

yi

xio

y

ex

ey

Piri

x

y P2

x

yi

xi

yPi

P1

d12xi'

y'

x'yi'

Ω

(a) (b) (c)

P3

φ213

Figure 2.1: 2D Cartesian coordinate system (a), definition of azimuth, angle and distance (b) and a 2D coordinatetransformation (c).

with < 𝐮, 𝐯 > the dot (inner) product of two vectors. See Figure 2.1b. The correspondingdistance ratio is defined as 𝑑12/𝑑13. Note that in surveying and mapping the azimuth, orbearing, is defined differently than in mathematics.

In land surveying the x­axis is usually (roughly) oriented in the East direction and the y­axisin the North direction. Therefore, the x­ and y­coordinates are also sometimes called Eastingand Northing. The azimuth, or bearing, is referred to the North direction. This can either bethe geographic North, magnetic North, or as is the case here, to the so­called grid North: thedirection given by the y­axis. The azimuth angle is defined as the angle of the vector 𝐫12 withthe North direction and is counted clockwise, i.e. for the azimuth a left­handed conventionis used, see Figure 2.1b. In mathematics the x­ and y­coordinate often called abscissa andordinate, and angles are counted counter­clockwise from the x­axis, with 𝜃12 = arctan 𝑦2−𝑦1

𝑥2−𝑥1following the mathematical text book definition of tangent. Thus 𝛼12 = 𝜋/2 − 𝜃12 in radians,or 𝛼12 = 90∘ − 𝜃12 when expressed in degrees.

Another possibility for describing the position of a point 𝑃𝑖 in a 2D Cartesian coordinatesystem is by its polar coordinates, which are the azimuth 𝛼𝑜𝑖 and distance 𝑑𝑜𝑖 to the point𝑃𝑖 from the origin of the coordinate system. For many types of surveying instruments andmeasurements it is often convenient to make use of polar coordinates. For instance, with atachymeter the distance and direction measurements in the horizontal plane are polar coordi­nates3 in a 2D local coordinate system, with origin in the instrument and y­axis in an arbitrary,yet to be determined, direction (representing the zero reading of the instrument).

2.2. 2D coordinate transformations2.2.1. Shape preserving transformationsThe 2D Cartesian coordinate system is defined by the origin of the axis, the direction of one ofthe axis (the second axis is orthogonal to the first) and the scale (the same for both axis). Thisbecomes immediately clear when a second Cartesian coordinate system is considered with axis𝑥′ and 𝑦′, see Figure 2.1c. The coordinates (𝑥′𝑖 ,𝑦′𝑖 ) for point 𝑃𝑖 in the new coordinate systemare related to the coordinates (𝑥𝑖,𝑦𝑖) in the original system through a rotation with rotationangle Ω, a scale change by a scale factor 𝑠 and a translation by two origin shift parameters

3In 3D, these become spherical coordinates, which is what a tachymeter measures.

2.3. Realization of 2D coordinate systems 5

(𝑡𝑥,𝑡𝑦) via a so–called (2D) similarity transformation,

( 𝑥′𝑖𝑦′𝑖) = 𝑠 ( cosΩ sinΩ

− sinΩ cosΩ )( 𝑥𝑖𝑦𝑖) + ( 𝑡𝑥𝑡𝑦

) . (2.4)

The transformation is a so­called conformal transformation whereby angles ∠𝐴𝑂𝐵 betweenpoints 𝐴, 𝑂 and 𝐵 are preserved, i.e. angles are not changed by the transformation. Thisalso means that shapes are preserved. A conformal transformation is therefore also knownas a similarity transformation or 2D Helmert transformation. Distances are not necessarilypreserved in a conformal or similarity transformation, unless the scale factor 𝑠 is one, butratios of distances 𝑑𝑂𝐴/𝑑𝑂𝐵 between three points are preserved.

The transformation involves four parameters: two translations 𝑡𝑥 and 𝑡𝑦, a rotation Ω,and, a scale factor 𝑠. This means that any 2D Cartesian coordinate system is uniquely definedby four parameters. Note that translation, rotation and scale only describe relations betweencoordinate systems, so there is always one coordinate system that is used as a starting point.Translation, rotation and scale change are relative concepts. However, a 2D Cartesian coor­dinate system can also be defined uniquely by assigning coordinates for (at least) two points.

In the special case that the scale factor 𝑠 is unity (𝑠 = 1) both angles and distances are pre­served in the transformation. This is called a congruence transformation. The transformationinvolves 3 instead of 4 parameters: two translations 𝑡𝑥 and 𝑡𝑦, and a rotation Ω. In this case,a 2D Cartesian coordinate system is defined either by (i) the three transformation parameterswith respect to another 2D coordinates system, or (ii) by assigning three coordinates for (atleast) two points.

2.2.2. Affine and polynomial transformationsTwo other types of transformations, that do not preserve shape, are affine, and the moregeneral polynomial transformations.

An affine transformation involves a rotation, scale change separately in both x­ and y­direction, and a translation. It can be written as,

( 𝑥′𝑖𝑦′𝑖) = ( 𝑎 𝑏

𝑐 𝑑 )(𝑥𝑖𝑦𝑖) + ( 𝑡𝑥𝑡𝑦

) . (2.5)

with the 2­by­2 transformation matrix containing 4 different elements. Affine transformationsintroduce so­called sheering between the coordinate axis. Angles are not necessarily preservedin an affine transformation, but lines remain straight, and parallel lines remain parallel afteran affine transformation. Also ratios of distances between points lying on a straight line arepreserved.

A polynomial transformation is a non­linear transformation which involves quadratic andoften higher order terms of the coordinates. Polynomial transformations are also given byEq. 2.5, but the transformation parameters are actually functions of the coordinates them­selves. This means that straight lines, and shapes, are not necessarily preserved. Polynomialtransformations are sometimes used as approximate transformations to match satelllite andaerial imagery onto a 2D Cartesian coordinate system, or as approximate transformation forgrid coordinates between two map projections, or to handle non­linear distortions in scannedhistorical maps.

2.3. Realization of 2D coordinate systemsAssigning coordinates for two points uniquely defines a 2D Cartesian system: two pointsrepresent four parameters (coordinates) from which the position, orientation and scale of the

6 2. 2D Cartesian coordinate systems

Figure 2.2: Two dimensional survey network with 4 angle measurements and 8 distance measurements.

axis can be constructed. The distance between two points defines the scale, from the azimuthbetween two points follows the orientation of the y­axis, and the coordinates themselvesdefine where the origin is.

This means that a coordinate reference system, when no pre­surveyed points are avail­able, can be established and realized by selecting a number of points in the field and assigningcoordinates to them. In its simplest form you could stake out a marker, assign this markerthe coordinates (0,0), stake out a second marker and assign this the coordinates (0,1), whichdefines the y­axis and length scale. But you also could have assigned different coordinates,thereby defining a different reference frame. Instead of assigning the rather arbitrary value1 for the y­coordinates of the second point, you could also have used a measured distancebetween the two points involved in the definition. This implies that the scale of our freshlydefined coordinate system is determined by the scale of the measuring device (e.g. a tapemeasure) and also any measurement errors that were made in these measurements are in­cluded in the definition of scale.

All this works well for defining a local coordinate system, but what about a national system,or that of a neighboring or previously realized project? In order to access any other systemyou should include at least two (observable) points for which coordinates in the other systemare known. These could be points that have been established by other organizations, such asa Cadaster or mapping agency which publishes the coordinates of many reference markers.It could also be points you have established yourself using for instance GPS measurements.

2.4. Worked out examplesBy means of two worked out examples we will show how a 2D coordinate system is realizedin a practical way and how this is related to linear algebra.

2.4.1. 2D coordinate system definitionIn this simple example, we show how to assign coordinates to points in the terrain, in order toestablish a coordinate system. We will — in a practical way — define the position and orienta­tion of a local 2D survey network. The scale is already implied by the distance measurements(in this example).

Figure 2.2 shows a simple survey network, with 5 points. Between these points, angleand distance measurements have been taken. When the coordinates of points 1 and 2 aregiven (e.g. as a result of an earlier survey), then these angle and distance measurements canbe used (and are sufficient) to determine the coordinates of points 3, 4 and 5, for instancethrough least­squares parameter estimation.

2.4. Worked out examples 7

Figure 2.3: Two dimensional simple survey network with 4 angle measurements and 6 distance measurements.

What now, if no coordinates are available a­priori? Then you have to choose some co­ordinates yourself, in order to establish a so­called local network. But, you have to make aconsiderate choice ­ you cannot just assign coordinate values to some random points. Forinstance, if we would assign coordinates to points 1, 2 and 3 (in an arbitrary way), we maycause deformations and distortions of the network — i.e. the coordinates of those points maythen not match (at all) the actually observed angles and distances!

The considerate choice requires you to analyse the geometry of the network. As statedin section 2.3, a 2D Cartesian coordinate system is uniquely defined by four parameters: thescale 𝑠, orientation Ω, and translations 𝑡𝑥 and 𝑡𝑦. The scale is set, in this case, by the distancemeasurements. What remains to be fixed are the origin and the orientation.

A geometric network, or construction, with angles and distances, like the one in Figure 2.2provides shape and scale, but not (absolute) position, nor orientation. You can shift thenetwork (in two directions), while the angles and distances between the points stay exactlythe same, and also, you can rotate the network, without altering angles and distances. Thereare still three degrees of freedom. Distances and angles are invariant against translation androtation. Or, to turn this around, angle and distance measurements lack information abouttranslation and rotation! Hence, you have to supply this!

In this example one could fix the coordinates of point 2, for instance, simply setting it to bethe origin (𝑥2, 𝑦2) = (0, 0) (this fixes two degrees of freedom). And, one could set point 5 ex­actly along the positive x­axis, hence setting its y­coordinate to zero (this fixes the last degreeof freedom). The coordinates of point 5 then become (𝑥5, 𝑦5) = (𝑙25, 0), where the measureddistance 𝑙25 is used for the x­coordinate. The distance and angle measurements pose a geo­metric defect with three degrees of freedom in a 2D­network. Hence, three coordinates shallbe fixed (no more, no less).

2.4.2. Algebraic analysisThe previous example provides a practical ‘recipe’ how to establish a 2D coordinate system.In the following we present, by means of a similar simple example, an algebraic analysis ofthis geometric defect.

The network is shown in Figure 2.3. This artificial network, being just a square, allows fora convenient and simple algebraic analysis. There are four angle measurements 𝛼314, 𝛼312,𝛼123, and 𝛼124. There are six distance measurements 𝑙12, 𝑙13, 𝑙14,𝑙23,𝑙24, and 𝑙34. Forgettingabout measurement errors, one can link these measurements to the (unknown) coordinates,by the following system of equations

𝐲 = 𝐀𝐱 (2.6)

8 2. 2D Cartesian coordinate systems

where the observations are in vector 𝐲 on the left, the unknown parameters (the coordinates)in vector 𝐱 on the right, and matrix 𝐀 relating the two. Often the relation is non­linear, whichis consequently approximated with a linearized relation, where we work with increments ofobservations and parameters (indicated by the Δ­symbol). Further details can be found in thePrimer on Mathematical Geodesy (in particular chapters 4 and 5). For the artificial geometryin Figure 2.3, this system of equations becomes:

⎛⎜⎜⎜⎜⎜⎜⎜

⎝

Δ𝛼314Δ𝛼312Δ𝛼123Δ𝛼124Δ𝑙12Δ𝑙13Δ𝑙14Δ𝑙23Δ𝑙24Δ𝑙34

⎞⎟⎟⎟⎟⎟⎟⎟

⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

10200

10200 0 0 − 10

100 0 10200 − 10

20010100

10100 0 − 10

100 − 10100 0 0 0

0 − 10100 − 10

20010200

10200

10200 0 0

0 − 10100 − 10

10010100 0 0 10

100 0−1 0 1 0 0 0 0 00 −1 0 0 0 1 0 0−√22 −√22 0 0 0 0 √2

2√22

0 0 √22 −√22 −√22

√22 0 0

0 0 0 −1 0 0 0 10 0 0 0 −1 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎛⎜⎜⎜⎜⎜

⎝

Δ𝑥1Δ𝑦1Δ𝑥2Δ𝑦2Δ𝑥3Δ𝑦3Δ𝑥4Δ𝑦4

⎞⎟⎟⎟⎟⎟

⎠

(2.7)

There are𝑚 = 4+6 = 10 observations, hence 𝐲 is a 10x1­vector, and there are 𝑛 = 8 unknownparameters in vector 𝐱. Consequently, matrix 𝐀 has dimensions 10x8. The rank of matrix 𝐀is however only 5, not 8. This is the algebraic indication that the measurements leave threedegrees of freedom. The null­space of matrix 𝐀 is not empty. Instead, in this example, thenull­space of matrix 𝐀 can be spanned by the following three (linearly independent) vectors:

𝐯1 =

⎛⎜⎜⎜⎜⎜

⎝

10101010

⎞⎟⎟⎟⎟⎟

⎠

; 𝐯2 =

⎛⎜⎜⎜⎜⎜

⎝

01010101

⎞⎟⎟⎟⎟⎟

⎠

; 𝐯3 =

⎛⎜⎜⎜⎜⎜

⎝

000−1101−1

⎞⎟⎟⎟⎟⎟

⎠

(2.8)

These three vectors can be stored together, in 8x3 matrix 𝐕, with 𝐕 = (𝐯1, 𝐯2, 𝐯3), for whichholds 𝐀𝐕 = 𝟎. The columns of matrix 𝐕 provide a basis for the null­space of matrix 𝐀.

Now suppose that 𝐱 is a solution to (2.6), then 𝐱′ = 𝐱 + 𝐕𝜷, with 3x1­vector 𝜷 =(𝛽1, 𝛽2, 𝛽3)𝑇, is also a solution, namely

𝐲 = 𝐀𝐱′ = 𝐀𝐱 + 𝐀𝐕𝜷 = 𝐀𝐱 (2.9)

Or, changing the coordinates of the points, in some particular way as imposed by matrix 𝐕,does not change the observations. Or, the other way around, based on a set of observations,you cannot tell the difference between 𝐱 and 𝐱′. The null­space of matrix 𝐀 being not empty,causes that there is left a certain degree of freedom in the solution.

The vectors 𝐯1, 𝐯2 and 𝐯3 can be easily interpreted in this example, see Figure 2.4. Vector𝐯1 implies an offset to the x­coordinates of all points in the network, meaning that translationparameter 𝑡𝑥 is undefined. Vector 𝐯2 implies an offset to the y­coordinates of all points,meaning that 𝑡𝑦 is undefined. And eventually, vector 𝐯3 implies a rotation of the networkabout point 1.

2.5. Problems and exercises 9

Figure 2.4: Interpretation of the null­space of matrix 𝐀, vector 𝐯1 at left, vector 𝐯2 in the middle, and vector 𝐯3 atright.

Applying the earlier practical ‘recipe’ would cause us to fix the coordinates of point 1 to theorigin (𝑥1, 𝑦1) = (0, 0), and to set the y­coordinate of point 2 to zero, i.e. (𝑥2, 𝑦2) = (𝑙12, 0).Three coordinates have been fixed, and consequently they can be removed from vector 𝐱.Correspondingly, the first, second, and fourth column of matrix 𝐀 has to be removed as well.

The interested reader is encouraged to verify that the resulting/reduced matrix 𝐀, withdimensions 10x5, has full rank, equal to 5, and an empty null­space. This means that, basedon the available measurements, the remaining 5 parameters (coordinates) can be determined,smoothly, for instance through least­squares estimation. The origin, the scale and the orien­tation of the 2D Cartesian coordinate system have been fixed.

2.5. Problems and exercisesQuestion 1 Two surveyors measure the facade of a building: points A and B. They both useEuclidean geometry in the local horizontal plane, but they adopt a different coordinate system,see the figure

The coordinates of point A and B in the blue system read (𝑥𝐴, 𝑦𝐴) = (2, 1), and (𝑥𝐵 , 𝑦𝐵) = (5, 1),and in the red system (𝑥′𝐴, 𝑦′𝐴) = (4√2,−√2), and (𝑥′𝐵 , 𝑦′𝐵) = (7√2,−4√2). The coordinatesin the two systems are related through a 2–D similarity transformation. Determine, based onthe coordinates given for the two points, the transformation parameters, i.e. scale factor 𝑠,rotation angle Ω, and translations 𝑡𝑥, 𝑡𝑦.Answer 1 A clever approach to solving this problem is using Eq. (2.4) on coordinate differ­ences

( 𝑥′𝐵 − 𝑥′𝐴𝑦′𝐵 − 𝑦′𝐵

) = 𝑠 ( cosΩ sinΩ− sinΩ cosΩ )( 𝑥𝐵 − 𝑥𝐴𝑦𝐵 − 𝑦𝐴

)

as the translation parameters cancel. Setting 𝑠 cosΩ = 𝑝 and 𝑠 sinΩ = 𝑞, we obtain

( 3√2−3√2 ) = (

𝑝 𝑞−𝑞 𝑝 )(

30 )

10 2. 2D Cartesian coordinate systems

from which we can easily solve 𝑝 and 𝑞. Doing so we find 𝑝 = √2 and 𝑞 = √2. From this wecan reconstruct that 𝑠 = √𝑝2 + 𝑞2 and Ω = arctan 𝑞

𝑝 , which gives 𝑠 = 2 and Ω =𝜋4 . Then

using again Eq. (2.4), but now for just one of the points, e.g. A, we have

( 4√2−√2 ) = 2(

12√2

12√2

−12√212√2

)( 21 ) + (𝑡𝑥𝑡𝑦)

from which we can solve the translation parameters as (𝑡𝑥 , 𝑡𝑦) = (√2, 0).

33D Cartesian coordinate systems

Three–dimensional (3D) coordinates systems are used to describe the position of objects in3D–space. In this chapter we discuss 3D Cartesian coordinate systems, before discussingspherical and ellipsoidal coordinate systems in Chapter 4. 3D Cartesian coordinate systemscan be considered a straightforward extension of 2D Cartesian coordinate systems by addinga third axis.

3.1. IntroductionThe position of an object in 3D space can be described in several ways. One of the moststraightforward ways is to give the position by three coordinates 𝑋, 𝑌, 𝑍 in a Cartesian coor­dinate system. See also Figure 3.1a. The coordinates are defined with respect to a referencepoint, or origin (with coordinates 0,0,0), which can be selected arbitrarily. For a global geocen­tric terrestrial coordinate system it is however convenient to choose the origin at the centerof the Earth. Further, the direction of one of the axis is chosen to coincide with the Earthrotation axis, while the other axis is based on a conventional definition of the zero meridian.The third axis completes the pair to make an orthogonal set of axes. The scale along the axesis simply tied to the SI definition of the meter (Appendix A). Capital letters 𝑋, 𝑌, 𝑍 are used forthe coordinates to set it apart from the 2D coordinate system in Chapter 2, but also becausethis is the usual notation for coordinates in a 3D global terrestrial coordinate system with theorigin at the center of the Earth.

Instead of a global geocentric terrestrial system also a local 3D Cartesian coordinate systemcan be defined, with the Y­axis pointing in the North direction, the Z­axis in the up direction,and the X­axis completing the pair and therefore pointing in the East direction, and with theorigin somewhere on the surface of the Earth. This type of system is referred to as topocentriccoordinate system and covered in Section 4.4. For the coordinates it is common to use thecapital letters 𝐸, 𝑁, 𝑈 (East, North, Up) instead of 𝑋, 𝑌, 𝑍.

In both examples, the geocentric as topocentric coordinates, the coordinate system issomehow tied to the Earth, but this is not necessary. An another common variant is wherethe coordinate system is tied to an instrument or sensor. Sometimes the third axis may bealigned to the direction of the gravity vector, as is typical for a theodolite or total station, butthe third axis may also be tied to the observing platform (boat, car, plane) and have a moreor less arbitrary orientation with respect to the Earth gravity field.

11

12 3. 3D Cartesian coordinate systems

Y

X

Z

ri

φ213

Xi

Zi

YiO

Pi

eX

eY

eZ

(a)

Y

X

Z

P1

(b)

P2

P3

ζ12

α12α13

ζ13 d12

α213

Figure 3.1: 3D Cartesian coordinate system (a) and definition of azimuth 𝛼12, horizontal angle 𝛼213, vertical angle𝜁12, angle 𝜑213 and distance 𝑑12 (b).

3.2. 3D Cartesian coordinatesThe 3D topocentric Cartesian coordinate system can be considered a straightforward exten­sion of a 2D Cartesian coordinate system. Just image in Figure 2.1 a z­axis from the originpointing outside the paper towards you. Coordinates, position vectors, distances, and anglesare defined in a similar fashion. The 3D position vector for a point 𝑃𝑖 with coordinates (𝑋𝑖,𝑌𝑖,𝑍𝑖)given by

𝐫𝑖 = 𝑋𝑖𝐞𝑋 + 𝑌𝑖𝐞𝑌 + 𝑍𝑖𝐞𝑍 , (3.1)

with 𝐞𝑋, 𝐞𝑌 and 𝐞𝑍 the unit vectors defining the axis of the Cartesian system. This is illustratedin Figure 3.1a. As shown in Figure 3.1b, the distance 𝑑12 between two points 𝑃1 and 𝑃2 is

𝑑12 = ‖𝐫2 − 𝐫1‖ = √(𝑋2 − 𝑋1)2 + (𝑌2 − 𝑌1)2 + (𝑍2 − 𝑍1)2 , (3.2)

and angle ∠𝑃2𝑃1𝑃3 between points 𝑃2, 𝑃1 and 𝑃3 is,

𝜑213 = ∠𝑃2𝑃1𝑃3 = arccos< 𝐫2 − 𝐫1, 𝐫3 − 𝐫1 >‖𝐫2 − 𝐫1‖ ‖𝐫3 − 𝐫1‖

(3.3)

with < 𝐮, 𝐯 > the dot (inner) product of two vectors. For a topocentric system, with the Z­axisin the up direction and Y­axis to the the North, the azimuth 𝛼12 and vertical angle 𝜁12 betweenpoints 𝑃1 and 𝑃2 can be defined as,

𝛼12 = arctan𝑋2 − 𝑋1𝑌2 − 𝑌1

𝜁12 = arctan𝑍2 − 𝑍1

√(𝑋2 − 𝑋1)2 + (𝑌2 − 𝑌1)2(3.4)

See also Figure 3.1b. For practical computations the arctan in Eq. (3.4) should be replacedwith the atan2 (𝑋2 − 𝑋1, 𝑌2 − 𝑌1) function in order to obtain the right quadrant for the azimuth𝛼12. Note that this definition only makes sense for topocentric systems where the Z­axis isoriented in the up direction and Y­axis to the North. Eq. (3.4) cannot be used for globalgeocentric terrestrial coordinate systems. The angle 𝜑213 of Eq. (3.3) is not the same as thehorizontal angle 𝛼213 in Figure 3.1b. The horizontal angle is defined as 𝛼213 = 𝛼13 − 𝛼12.

3.3. 3D similarity transformations 13

Y’

Y

X’

Z=Z’

X

Y’=Y’’

Y

X’

X’’

Z’’

ZZ=Z’

X

Y’=Y’’

Y’’’

Y

X’’=X’’’

Z’’

Z=Z’

Z’’’

X

ΩZ

ΩY

ΩX

Figure 3.2: Definition of rotation angles for the 7­parameter similarity transformation. The figure on the leftshows a rotation Ω𝑧 about the Z­axis, the middle figure a rotation Ω𝑦 about the newly obtained Y’­axis, and thefigure on the right a rotation Ω𝑥 about the final X”­axis. The angles are positive for a counter­clockwise rotationwhen viewed along the axis towards the origin (right­handed rotation is positive) and defined to turn the sourcecoordinate system axes into the target system axes.

3.3. 3D similarity transformationsWe start this section with a brief overview of 3D coordinate transformations. Of the 3D trans­formations, the 3D similarity transformation, that preserves shape, is by far the most oftenused coordinate transformation for 3D coordinates, and the remainder of this section is de­voted to this important type of transformation.

3.3.1. Overview 3D coordinate transformationsThe affine transformation is the most general transformation which can be represented interms of linear algebra. The 3­by­3 matrix 𝐑 has nine different elements, implying rotation,scaling and so­called shearing, the latter meaning that a square is turned into a parallelogram(or, actually a cube into a parallelepiped).

(𝑋′𝑌′𝑍′) = (

𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑔 ℎ 𝑖

)⏝⏟⏝

𝐑

(𝑋𝑌𝑍) + (

𝑡𝑥𝑡𝑦𝑡𝑧) (3.5)

The 3D affine transformation is specified by a total of 12 parameters.The similarity transformation preserves the shape of objects. The 3­by­3 matrix 𝐑 now

implies only a rotation (or actually series of rotations). Matrix 𝐑 has 9 elements, but needs tosatisfy 3 orthogonality conditions and 3 orthonormality conditions (the rows are orthogonal,and they are all of unit length), and thereby only 3 degrees of freedom remain (3 rotationangles).

(𝑋′𝑌′𝑍′) = 𝜆𝐑(Ω𝑥 , Ω𝑦 , Ω𝑧) (

𝑋𝑌𝑍) + (

𝑡𝑥𝑡𝑦𝑡𝑧) (3.6)

The 3D similarity transformation is specified by a total of 7 parameters. In the sequel thescale parameter 𝜆, which often close to one, will be replaced by 1 + 𝜇.

The congruence transformation preserves the shape and size of objects. It is the so­called‘rigid body’ transformation. It is a special case of the similarity transformation, with the scale

14 3. 3D Cartesian coordinate systems

parameter fixed to one 𝜆 = 1 (or equivalently, 𝜇 = 0).

(𝑋′𝑌′𝑍′) = 𝐑(Ω𝑥 , Ω𝑦 , Ω𝑧) (

𝑋𝑌𝑍) + (

𝑡𝑥𝑡𝑦𝑡𝑧) (3.7)

The 3D congruence transformation is specified by a total of 6 parameters (3 for rotation, and3 for translation).

Of the three transformations, the 3D similarity transformation is by far the most often used3D coordinate transformation. It will be covered in more detail in the next subsections.

3.3.2. 7­parameter similarity transformationTo transform 3D Cartesian coordinates from a source to target coordinate system a 7­parametersimilarity transformation is used. The transformation consists of three translations (𝑡𝑥 , 𝑡𝑦 , 𝑡𝑧),three rotations (Ω𝑥 , Ω𝑦 , Ω𝑧) and a differential scale factor 𝜇,

(𝑋′𝑌′𝑍′) = (1 + 𝜇) ⋅ 𝐑(Ω𝑥 , Ω𝑦 , Ω𝑧) ⋅ (

𝑋𝑌𝑍) + (

𝑡𝑥𝑡𝑦𝑡𝑧) (3.8)

with (𝑋, 𝑌, 𝑍) the coordinates in the source coordinate system and (𝑋′, 𝑌′, 𝑍′) the coordinatesin the target coordinate system. The differential scale factor 𝜇 is often a small number andis sometimes expressed in parts­per­million (ppm), with 1ppm = 10−6. The scale factor(1 + 𝜇) is then close to one. The translation vector (𝑡𝑥 , 𝑡𝑦 , 𝑡𝑧) has to be added to the sourcecoordinates after rotation. The translation vector gives the coordinates of the origin of thesource coordinate system with respect to the target coordinate system. The rotation matrix𝐑(Ω𝑥 , Ω𝑦 , Ω𝑧) is defined as a sequence of so–called Euler rotations. A 3­2­1 series of Eulerrotations gives the rotation matrix

𝐑321(Ω𝑥 , Ω𝑦 , Ω𝑧) = 𝐑1(Ω𝑥) ⋅ 𝐑2(Ω𝑦) ⋅ 𝐑3(Ω𝑧) =

⎛⎜

⎝

cosΩ𝑧 cosΩ𝑦 sinΩ𝑧 cosΩ𝑦 − sinΩ𝑦cosΩ𝑧 sinΩ𝑦 sinΩ𝑥 − sinΩ𝑧 cosΩ𝑥 sinΩ𝑧 sinΩ𝑦 sinΩ𝑥 + cosΩ𝑧 cosΩ𝑥 cosΩ𝑦 sinΩ𝑥cosΩ𝑧 sinΩ𝑦 cosΩ𝑥 + sinΩ𝑧 sinΩ𝑥 sinΩ𝑧 sinΩ𝑦 cosΩ𝑥 − cosΩ𝑧 sinΩ𝑥 cosΩ𝑦 cosΩ𝑥

⎞⎟

⎠

(3.9)

with Ω𝑧, Ω𝑦 and Ω𝑥 the rotation angles around – and in that order – the z­, y­ and x­axisrespectively. The corresponding Euler rotation matrices, 𝐑𝑖(Ω𝑖), describing rotations aroundthe coordinate axis, are

𝐑3(Ω𝑧) = (cosΩ𝑧 sinΩ𝑧 0− sinΩ𝑧 cosΩ𝑧 0

0 0 1) , 𝐑2(Ω𝑦) = (

cosΩ𝑦 0 − sinΩ𝑦0 1 0

sinΩ𝑦 0 cosΩ𝑦) ,

𝐑1(Ω𝑥) = (1 0 00 cosΩ𝑥 sinΩ𝑥0 − sinΩ𝑥 cosΩ𝑥

)

(3.10)

whereby the right­handed rotation is positive, which is, when viewed along the axis towardsthe origin, a counter­clockwise rotation. See Figure 3.2. This sense of rotation is the same aswas used in the 2­dimensional case, see Eq. (2.4). Imagine in Figure 2.1c a z­axis pointingout of the paper, then the rotation matrix of Eq. (2.4) is essentially 𝐑3(Ω𝑧) (which does notchange the z­axis or z­coordinates) and the rotation angle Ω of Eq. (2.4) is actually Ω𝑧 in

3.3. 3D similarity transformations 15

the 3D­transformation. The rotation Ω𝑧 around the z­axis is actually a rotation of the x­axis(and also y­axis) by Ω𝑧. The complete rotation 𝐑321(Ω𝑥 , Ω𝑦 , Ω𝑧) is thus the product of first arotation around the z­axis, followed by a rotation around the new y­axis, and finally a rotationaround the then current x­axis.

Y

Z

X

ΩZ

ΩY

ΩX

Figure 3.3: Definition of infinitesimal small rotation angles Ω𝑥, Ω𝑦 and Ω𝑧 for the Helmert transformation.

Changing the order of the Euler rotations in Eq. (3.9) will result in a different equation forthe rotation matrix with different rotation angles. This is typical for Euler rotations. Revertingthe order of rotations in Eq. 3.9, gives a 1­2­3 sequence of Euler rotations with rotation matrix,

𝐑123(Ω′𝑥 , Ω′𝑦 , Ω′𝑧) = 𝐑3(Ω′𝑧) ⋅ 𝐑2(Ω′𝑦) ⋅ 𝐑1(Ω′𝑥) =

⎛⎜

⎝

cosΩ′𝑧 cosΩ′𝑦 cosΩ′𝑧 sinΩ′𝑦 sinΩ′𝑥 + sinΩ′𝑧 cosΩ′𝑥 − cosΩ′𝑧 sinΩ′𝑦 cosΩ′𝑥 + sinΩ′𝑧 sinΩ′𝑥− sinΩ′𝑧 cosΩ′𝑦 − sinΩ′𝑧 sinΩ′𝑦 sinΩ′𝑥 + cosΩ′𝑧 cosΩ′𝑥 sinΩ′𝑧 sinΩ′𝑦 cosΩ′𝑥 + cosΩ′𝑧 sinΩ′𝑥

sinΩ′𝑦 − cosΩ′𝑦 sinΩ′𝑥 cosΩ′𝑦 cosΩ′𝑥

⎞⎟

⎠

(3.11)

with Ω′𝑥, Ω′𝑦 and Ω′𝑧 the rotation angles around – and in that order – the x­, y­ and x­axisrespectively. The rotation matrix 𝐑123(Ω𝑥 , Ω𝑦 , Ω𝑧) is thus the product of first a rotation aroundthe x­axis, followed by a rotation around the new y­axis, and finally a rotation around the thencurrent z­axis. The rotation angles Ω′𝑥, Ω′𝑦 and Ω′𝑧 are different from the rotation angles Ω𝑥,Ω𝑦 and Ω𝑧 used in Eq. 3.9.

The rotation cannot be inverted by just changing the sign of the parameters (except forvery small angles). The inverse of the rotation matrix is 𝐑321(Ω𝑥 , Ω𝑦 , Ω𝑧)−1 = (𝐑1(Ω𝑥)⋅𝐑2(Ω𝑦)⋅𝐑3(Ω𝑧))−1 = 𝐑3(−Ω𝑧) ⋅ 𝐑2(−Ω𝑦) ⋅ 𝐑1(−Ω𝑥) = 𝐑123(−Ω𝑥 , −Ω𝑦 , −Ω𝑧). This is not the same aschanging the sign of the angles in Eq. 3.9, it also means changing the order of the rotations. Asthe rotation matrices are orthogonal matrices, the inverse of the rotation matrix is equal to thetranspose of the matrix. Thus we can write 𝐑321(Ω𝑥 , Ω𝑦 , Ω𝑧)𝑇 = (𝐑1(Ω𝑥) ⋅𝐑2(Ω𝑦) ⋅𝐑3(Ω𝑧))𝑇 =𝐑3(Ω𝑧)𝑇 ⋅ 𝐑2(Ω𝑦)𝑇 ⋅ 𝐑1(Ω𝑥)𝑇 = 𝐑3(−Ω𝑧) ⋅ 𝐑2(−Ω𝑦) ⋅ 𝐑1(−Ω𝑥) = 𝐑123(−Ω𝑥 , −Ω𝑦 , −Ω𝑧), or inshort, 𝐑321(Ω𝑥 , Ω𝑦 , Ω𝑧)𝑇 = 𝐑123(−Ω𝑥 , −Ω𝑦 , −Ω𝑧). Compare the terms in Eqs. 3.9 and 3.11,and you will see it is true. It means that to do the reverse rotation, we not only have to changethe sign of the rotation angles, but also need to revert the order of the rotations.

16 3. 3D Cartesian coordinate systems

3.3.3. 7–parameter Helmert (small angle) transformationIn the case when the rotation angles are very small, with cosΩ ≃ 1 and sinΩ ≃ Ω (with Ω inradians), the rotation matrix 𝐑(Ω𝑥 , Ω𝑦 , Ω𝑧) is then

𝐑(Ω𝑥 , Ω𝑦 , Ω𝑧) ≃ (1 Ω𝑧 −Ω𝑦−Ω𝑧 1 Ω𝑥Ω𝑦 −Ω𝑥 1

) (3.12)

with the rotation angles as defined in Figure 3.3. The 7­parameter similarity transformationof Eq. (3.8) in it’s simplified form is

(𝑋′𝑌′𝑍′) = (

𝑋𝑌𝑍) + (

𝑡𝑥𝑡𝑦𝑡𝑧) + (

𝜇 Ω𝑧 −Ω𝑦−Ω𝑧 𝜇 Ω𝑥Ω𝑦 −Ω𝑥 𝜇

) ⋅ (𝑋𝑌𝑍) (3.13)

This transformation is also known as the 7­parameter Helmert transformation (the 3­parameterHelmert transformation only includes the translation). This transformation is reversible: chang­ing the sign of the seven transformation parameters results in the inverse transformation.

The reader should be aware that often different conventions are used for the sign of therotation parameters. The convention that is used in this reader is that a positive rotation is acounter­clockwise rotation when viewed in the direction of the origin, and this convention isapplied to a rotation of the axis of the coordinate system. Other conventions define transfor­mations not based on rotation of the axis, but are based on rotations of the position vector,resulting in an opposite sign for the rotation angles or other signs for the small angle termsin the rotation matrices. It is always a good idea to check that the transformation formulasprovided together with published transformation parameters use the same sign­convention asthe software you are using.

3.3.4. 10–parameter Molodensky–Badekas transformation [*]The 7­parameter similarity transformation uses rotations about the origin of the source system.This may result in numerical problems for networks of points that are confined to small regionson the Earth surface, such as coordinates of a national reference system. In this case there willbe a high correlation between the translations and rotations in the derivation of the parametervalues for the standard 7­parameter transformation. Therefore, instead of rotations beingderived around the origin of the system which is near the geocenter, rotations are derivedaround a point somewhere within the domain of the network (e.g. in the middle of the area ofinterest on the Earth’s surface). For this type of transformation three additional parameters,the coordinates of the rotation point, are required to describe the transformation. Theseadditional parameters can be chosen freely, or by convention, and do not have the samerole in the derivation of parameter values for the other 7­parameters. The transformationessentially remains a 7­parameter transformation, with 7 degrees of freedom, although anextra 3 parameters are needed in the specification. The transformation formula is

(𝑋′𝑌′𝑍′) = (1 + 𝜇) ⋅ 𝐑(Ω′𝑥 , Ω′𝑦 , Ω′𝑧) ⋅ (

𝑋 − 𝑋0𝑌 − 𝑌0𝑍 − 𝑍0

) + (𝑡′𝑥𝑡′𝑦𝑡′𝑧) + (

𝑋0𝑌0𝑍0

) (3.14)

with (𝑋0, 𝑌0, 𝑍0) the coordinates of the selected rotation point. This transformation is notreversible in the sense that the same parameter values, with different signs, can be used forthe reverse transformation. This is because the coordinates for the rotation point are changedby the transformation. However, in practice sometimes the same coordinates are used, but

3.4. Realization of 3D coordinate systems 17

this results in cumulative errors after repeated transformations. Eq. (3.14) uses the same signconvention as Eq. (3.8). Note that, whereas many publications use for Eq. (3.8) the samesign convention as this reader, most publications use for Eq. (3.14) the opposite convention.You are warned.

3.4. Realization of 3D coordinate systemsThe 3D Cartesian coordinate system is defined by the origin of the axes, the direction oftwo axes (the third axis is orthogonal to the other two) and the scale, which is the samefor all axes. This becomes immediately clear when a second Cartesian coordinate system isconsidered with axes 𝑋′, 𝑌′ and 𝑍′. The coordinates (𝑋′𝑖,𝑌′𝑖 ,𝑍′𝑖) for point 𝑃𝑖 in the new coordinatesystem are related to the coordinates (𝑋𝑖,𝑌𝑖,𝑍𝑖) in the original system through a 7­parametersimilarity transformation, consisting of three rotations, three translations and a scale factor,see Section 3.3. This transformation is again a conformal (similarity) transformation, wherebyangles ∠𝐴𝑂𝐵 and distance ratio’s between points 𝐴, 𝑂 and 𝐵 are preserved, i.e. shapes arenot changed by the transformation. In the transformation 7 parameters are involved. Thismeans that any 3D Cartesian coordinate system is uniquely defined by 7 parameters. Notethat again translation, rotation and scale only describe relations between coordinate systems,which means that there is always one coordinate system that is used as a starting point.

However, a 3D Cartesian coordinate system can also be defined uniquely by assigning co­ordinates for (at least) three points. Assigning coordinates for two points uniquely definessix degrees of freedom, which leaves one degree of freedom (a rotation) which needs to beresolved by one coordinate of a third point (although one coordinate is sufficient for the defi­nition, approximate values for the other two coordinates are needed for numerical reasons).This means that a 3D­coordinate reference system can be realized by selecting at least 3points and assigning at least 7 coordinates to them.

Using 7 coordinates for 3 points is just enough to define a 3D­coordinate reference system.When using more than 7 coordinates and 3 points to define the coordinates there is a seriousrisk of introducing distortions in the coordinates. For example, suppose we have 3 points,each with 3 coordinates. Suppose we use the coordinates of the first two points and theZ­coordinate of the third point to define the coordinate system, then in general the X and Ycoordinates of the third point will not match the given coordinates, unless by coincidence. Thesame is true if four or more points are ‘given‘. The only proper way to handle such a situation isto set up a system of equations with a 3D­similarity transformation and then minimize in least–squares sense the differences between the given and computed coordinates. In more technicalterms this is known as an S­transformation. In this way, using more than 7 coordinatesfor 3 points, has the important advantage of added redundancy in practical computations andbecoming less sensitive to outliers, especially in combination with statistical testing, withoutintroducing distortions in the coordinate network. This is called a free–network.

It is also possible that we want to do an over­determined connection to given coordinates.In this case the resulting coordinates for the connection points with be the same as the givenvalues, but the original network of coordinates will be distorted. This can also be done in aweighted sense, whereby weights or a co–variance matrix is assigned to the given coordinates,and the network of coordinates is fitted in least–squares sense to the given coordinates. Thisresults in an over–determined network of coordinates.

3.5. Problems and exercisesQuestion 1 Coordinate system I is related to coordinate system II through a rotation (counter­clockwise) about the Z­axis over 90 degrees. Both systems are three­dimensional Cartesiancoordinate systems. Compute the rotation matrix for transforming coordinates given with

18 3. 3D Cartesian coordinate systems

respect to system I, into coordinates with respect to system II.Answer 1 The 3x3 rotation matrix is given by Eq. (3.10). The angle Ω𝑧 = 90 degrees. Hence,the matrix becomes

(0 1 0−1 0 00 0 1

) .

So, a point on the Y­axis in source system I (e.g. with coordinates (0, 1, 0)), becomes a pointon the X­axis in targete system II (e.g. with coordinates (1, 0, 0)).

4Spherical and ellipsoidal coordinate

systems

Although straightforward, Cartesian coordinates are not very convenient for representing po­sitions on the surface of the Earth. Take a global terrestrial coordinate system, with origin atthe center of mass of the Earth, the Z­axis is coinciding with the Earth rotation axis, the X­axisis based on a conventional definition of the zero meridian and the Y­axis completing the pairto make an orthogonal set of axis, and scale tied to the SI definition of the meter. Consideringthat the Earth radius is about 6, 400 km, then to represent positions on the surface of theEarth 7 digits (before the decimal point) would be needed for each of the three coordinates.For representing positions on the surface of the Earth it is actually more convenient to usecurvilinear coordinates defined on a sphere or ellipsoid approximating the Earth’s surface.

4.1. Spherical coordinates (geocentric coordinates)For example, assuming a sphere with radius 𝑅 approximating the Earth surface, sphericalcoordinates 𝜓, 𝜆 and 𝑟 (with 𝑟 = 𝑅+ℎ′, and 𝑅 = 6, 371 km the mean radius of the Earth) canbe defined, see Figure 4.1. The relationship between Cartesian and spherical coordinates isgiven by,

𝑋 = 𝑟 cos𝜓 cos 𝜆𝑌 = 𝑟 cos𝜓 sin 𝜆𝑍 = 𝑟 sin𝜓

(4.1)

The inverse relationship is given by,

𝜓 = arctan( 𝑍√𝑋2 + 𝑌2)

)

𝜆 = arctan(𝑌𝑋)

𝑟 = √𝑋2 + 𝑌2 + 𝑍2)

(4.2)

The spherical coordinates 𝜓 and 𝜆 can be used to represent positions on the sphere. In thiscase the sphere is a coordinate surface (surface on which one of the coordinates is constant),with 𝜓 the geocentric latitude and 𝜆 the longitude of the point. In Eqs. (4.1) and (4.2) weabstained from using the expression 𝑅 +ℎ′, with 𝑅 the radius of the sphere and ℎ′ the heightabove the reference sphere. In particular, we abstained from using ℎ′ as a third coordinate.

19

20 4. Spherical and ellipsoidal coordinate systems

Instead we used the geocentric radius or distance 𝑟 of the point. This is because the sphereis not a very good approximation of the surface of the Earth and heights defined with respectto the sphere are meaningless (e.g. Mount Everest would have a height of 20 km, and theocean surface in the Arctic a height of −10 km).

Y

λ

X

Z

r

ψ

Figure 4.1: Spherical coordinates 𝜓, 𝜆, 𝑟 and Cartesian coordinates 𝑋, 𝑌, 𝑍.

4.2. Geographic coordinates (ellipsoidal coordinates)As shown by Newton (Principia, 1687) a rotating self­gravitating fluid body in equilibriumtakes the form of an oblate ellipsoid. The oblate ellipsoid, or simply ellipsoid, is a much betterapproximation for the shape of the Earth than a sphere. An ellipsoid is the three dimensionalsurface generated by the rotation of an ellipse about its shorter axis. Two parameters arerequired to describe the shape of an ellipsoid. One is invariably the equatorial radius, whichis the semi­major axis, 𝑎. The other parameter is either the polar radius or semi­minor axis,𝑏, or the flattening, 𝑓, or the eccentricity, 𝑒. They are related by

𝑓 = 𝑎 − 𝑏𝑎 , 𝑒2 = 2𝑓 − 𝑓2 = 𝑎2 − 𝑏2

𝑎2 , 𝑏 = 𝑎(1 − 𝑓) = 𝑎√1 − 𝑒2 (4.3)

For the Earth the semi­major axis 𝑎 is about 6, 378 km and semi­minor axis 𝑏 about 6, 357 km, a21 km difference. The flattening is of the order 1/300, which is indistinguishable in illustrationsif drawn to scale (illustrations, such as in this text, always exaggerate the flattening). Also,since 𝑓 is a very small number, instead of 𝑓 often the inverse flattening 1/𝑓 is given.

The position of a point with respect to an ellipsoid is given in terms of geographic or geode­tic latitude 𝜑, longitude 𝜆 and height ℎ above the ellipsoid, see Figure 4.2. The relationshipbetween Cartesian and geographic coordinates is given by,

𝑋 = ( + ℎ) cos𝜑 cos 𝜆𝑌 = ( + ℎ) cos𝜑 sin 𝜆𝑍 = ((1 − 𝑒2) + ℎ) sin𝜑

(4.4)

4.2. Geographic coordinates (ellipsoidal coordinates) 21

Y

λ

X

Z

h

φ

Figure 4.2: Ellipsoidal and Cartesian coordinates. The ellipsoidal latitude 𝜑 is also known as geodetic or geographiclatitude. The ellipsoidal coordinates 𝜑 and 𝜆 are also called geographic coordinates.

a

bN

h

φψ

r

λ

Figure 4.3: Ellipsoidal, geodetic or geographic latitude 𝜑, geocentric (or spherical) latitude 𝜓, radius of curvature = (𝜑), radius 𝑟, ellipsoidal height ℎ, semi­major axis 𝑎 and semi­minor axis 𝑏 of the ellipsoid.

The inverse relationship is given by,

𝜑 = arctan(𝑍 + 𝑒2 sin𝜑

√𝑋2 + 𝑌2)

𝜆 = arctan(𝑌𝑋)

ℎ = √𝑋2 + 𝑌2cos𝜑 −

(4.5)

in Eqs. (4.4) and (4.5) is the radius of curvature in the prime vertical, as shown in Figure 4.3.The radius of curvature for an ellipsoid depends on the location on the ellipsoid. It is a

function of the geographic latitude and is different in East­West and North­South direction.Theyare called respectively radius of curvature in the prime vertical, = (𝜑), and the radius ofcurvature in the meridian, = (𝜑), These two radii are not the same as the physical radius,the distance from the center of the Earth to the ellipsoid. This is different from a sphere,where all three radii are the same, and have a single value 𝑅. The radius of curvature in the

22 4. Spherical and ellipsoidal coordinate systems

prime vertical, = (𝜑), and the radius of curvature in the meridian, = (𝜑), for anellipsoid are

(𝜑) = 𝑎

√1 − 𝑒2 sin2 𝜑

(𝜑) = 𝑎(1 − 𝑒2)(1 − 𝑒2 sin2 𝜑)3/2

(4.6)

with radius of curvature normal to . On the equator the radius of curvature in East­West isequal to the semi­major axis 𝑎, with (0∘) = 𝑎, while the radius of curvature in North­South issmaller than the semi­minor axis, with (0∘) = 𝑎(1−𝑒2) = 𝑏(1−𝑓) = 𝑏2/𝑎. On the poles theradius of curvature (±90∘) = (±90∘) = 𝑎/√(1 − 𝑒2) = 𝑎2/𝑏 is larger than the semi­majoraxis 𝑎, see Figure 4.4.

0 10 20 30 40 50 60 70 80 906330

6340

6350

6360

6370

6380

6390

6400

Latitude [deg]

Rad

ius

[km

]

M

a

b

N

Figure 4.4: Radius of curvature (𝜑) and (𝜑) as function of latitude 𝜑. The dashed lines represent the semi­major axis 𝑎 and semi­minor axis 𝑏.

The radii of curvature play a role in the conversion of small differences in latitude andlongitude into linear distances on the surface of the Earth. If 𝑑𝜑 is the differential latitude inradians, and 𝑑𝜆 the differential longitude in radians, then

𝑑𝑁 = ((𝜑) + ℎ) 𝑑𝜑𝑑𝐸 = ((𝜑) + ℎ) cos𝜑 𝑑𝜆 (4.7)

with 𝑑𝑁 the differential distance in North­South (latitude) direction, with positive direction tothe North, and 𝑑𝐸 the differential distance in East­West (longitude) direction, with (𝜑) and(𝜑) the meridian radius of curvature and radius of curvature in the prime vertical as givenby Eq. (4.6) and Figure 4.4. Both 𝑑𝑁 and 𝑑𝐸 are in units of meters and are often referred toas Northing and Easting. The relations in Eq. (4.7) come in very handy if you wish to expresssmall differences in latitude and longitude in units of meters. This happens for instance whenyou have latitude and longitude for two nearby points, but instead of a latitude and longitudedifferences in angular units, you are more interested to have the difference in meters. It is alsovery useful to convert for instance standard deviations in angular units to standard deviationsin meters. For a first approximation, e..g. when differences in latitude and longitude are smallor when accuracy does not matter, (𝜑)+ℎ and (𝜑)+ℎ in Eq. (4.7) can be replaced simplyby the radius 𝑅 of the spherical Earth.

4.3. Astronomical latitude and longitude [*] 23

The geographic latitude in Eq. (4.5) must be computed by an iteration process as thegeographic latitude 𝜑 appears both in the left and right hand side of the equation. Also notethat the radius of curvature in Eq. (4.6), which is a function of the geographic latitude (thatstill needs to be computed by Eq. (4.5)), can be computed by the same iteration process. Theiterative procedure, whereby in the first iteration 𝑁′ = sin𝜑 is approximated by 𝑍, reads

𝑁′0 = 𝑍for 𝑖 = 1, 2, …

𝜑𝑖 = arctan(𝑍 + 𝑒2𝑁′𝑖−1

√𝑋2 + 𝑌2)

𝑖 =𝑎

√1 − 𝑒2 sin2 𝜑𝑖𝑁′𝑖 = 𝑖 sin𝜑𝑖

(4.8)

Usually four iterations are sufficient. For points near the surface of the Earth 𝜑 can also becomputed using a direct method of B.R. Bowring (Survey Review, 181, July 1976, p. 323­327),

𝜑 = arctan( 𝑍 + 𝑒′2𝑏 sin3 𝜇√𝑋2 + 𝑌2 − 𝑒2𝑎 cos3 𝜇

) (4.9)

with 𝑒′2, also called the second eccentricity, and 𝜇 given by

𝑒′2 = 𝑒21 − 𝑒2 =

𝑎2 − 𝑏2𝑏2

𝜇 = arctan( 𝑎𝑍𝑏√𝑋2 + 𝑌2

)(4.10)

The error introduced by this method is negligible for points between −5 and 10 km from theEarth surface, and, certainly much smaller than the error after four iterations with the iterativemethod.

The relation between the geocentric latitude 𝜓 and the geodetic (or geographic) latitude𝜑 for a point on the surface of the Earth, see Figure 4.3, is

𝜓(𝜑) = arctan((1 − 𝑒2) + ℎ

+ ℎ tan𝜑) ≃ arctan((1 − 𝑒2) tan𝜑) | ℎ ≪ (4.11)

The geodetic and geocentric latitudes are equal at the equator and poles. The maximumdifference of 𝜑 − 𝜓 is approximately 11.5 minutes of arc1 at a geodetic latitude of 45∘5′.The geocentric and geodetic longitude are always the same. However, it is important not toconfuse geocentric and geodetic latitude, which otherwise could result in an error in positionof up to 20 km.

4.3. Astronomical latitude and longitude [*]The normal, or vertical, to the ellipsoidal surface is the coordinate line that corresponds to ℎand (𝜑). The ellipsoidal normal at the observation point (𝜑, 𝜆) is given by the unit directionvector

= (cos𝜑 cos 𝜆cos𝜑 sin 𝜆

sin𝜑) (4.12)

11 minute of arc is 1/60 of a degree; so 11.5 minutes of arc is equal to 11.5/60 ≃ 0.19∘.

24 4. Spherical and ellipsoidal coordinate systems

The ellipsoidal normal does not pass through the centre of the ellipsoid, except at the equatorand at the poles. In general, the ellipsoidal normal does not coincide with the true vertical,𝐧, or plumb­line (in Dutch: schietlood) given by the direction of the local gravity field, 𝐠,at that point. Gravity is the resultant of the gravitational acceleration and the centrifugalacceleration at that point, see also Chapter 7. The direction of the true vertical 𝐧 is given bythe astronomical latitude 𝜙 and longitude Λ,

𝐧 = −𝐠𝑔 = (cos𝜙 cosΛcos𝜙 sinΛ

sin𝜙) (4.13)

with 𝑔 = ‖𝐠‖. The astronomical latitude and longitude can be determined through (zenith)measurements to the stars. The astronomical latitude 𝜙 is the angle between the equatorialplane and the true vertical at a point on the surface; the ellipsoidal, geodetic or geographiclatitude 𝜑 is the angle between the equatorial plane and the ellipsoidal normal. A similardistinction exists for the astronomical longitude Λ and ellipsoidal longitude 𝜆. The ellipsoidis a purely geometric shape, but astronomical latitude and longitude are driven by physics,namely the direction of gravity.

The angle between the directions of the ellipsoidal normal and true vertical at a point iscalled the deflection of the vertical. The deflection of the vertical is divided in two components,defined as,

𝜉 = 𝜙 − 𝜑𝜂 = (Λ − 𝜆) cos𝜑 (4.14)

Astronomical latitude 𝜙 and longitude Λ are obtained from astronomical observations to starswhose positions (declination 𝛿 and right ascension 𝛼 ) in a Celestial reference system areaccurately known, or from gravity observations using gravimeters2. The deflection of thevertical is usually only a few seconds of arc, whereby the largest values occur in mountainousareas and in areas with large gravity anomalies.

4.4. Topocentric coordinates, azimuth and zenith angle [*]It is not always convenient to use Cartesian coordinates in a global reference system with theorigin in the center of mass of the Earth. Sometimes it is more convenient to choose the originin a point on or near the surface of the Earth, and define the coordinate axis with respect tothe local vertical and geographic North. This type of 3D Cartesian coordinate system is called alocal topocentric coordinate system and it’s coordinates are called topocentric coordinates. InFigure 4.5 the origin of the local topocentric system is the (observation) point A with geographiccoordinates (𝜑, 𝜆, ℎ). The vectors 𝐸, 𝑁 and form the three axis of a right­handed localtopocentric system centered at A, with z­axis along the normal of the ellipsoid , x­axis 𝐸orthogonal to the plane of the meridian and positive to the East, and y­axis 𝑁 = × 𝐸 inthe plane of the meridian completing the topocentric system. The coordinates in the localtopocentric system are denoted by 𝐸 (East), 𝑁 (North) and 𝑈 (Up). The system itself issometimes also called a East­North­Up (ENU) coordinate system.

2Astronomical latitude is not to be confused with declination, the coordinate astronomers used in a similar way todescribe the locations of stars north/south of the celestial equator, nor with ecliptic latitude, the coordinate thatastronomers use to describe the locations of stars north/south of the ecliptic.

4.4. Topocentric coordinates, azimuth and zenith angle [*] 25

Y

λ

X

Z

h

φ

A

Bα

ζ_n_

eN_eE

Figure 4.5: Local right­handed topocentric system in point A, with ellipsoidal coordinates (𝜑, 𝜆, ℎ), with the localazimuth 𝛼 and zenith angle 𝜁 for the direction 𝐴𝐵. The vertical (ellipsoidal normal vector) is the z­axis of thelocal right­handed system , 𝐸 is the x­axis and is orthogonal to the plane of the meridian and positive to the East,and 𝑁 = × 𝐸 ,in the plane of the meridian, is the y­axis completing the topocentric system.

The relation between the differential coordinates (Δ𝑋, Δ𝑌, Δ𝑍) and (Δ𝐸, Δ𝑁, Δ𝑈) is,

(Δ𝑋Δ𝑌Δ𝑍

) = (− sin 𝜆 − sin𝜑 cos 𝜆 cos𝜑 cos 𝜆cos 𝜆 − sin𝜑 sin 𝜆 cos𝜑 sin 𝜆0 cos𝜑 sin𝜑

)(Δ𝐸Δ𝑁Δ𝑈

)

= ( 𝐸 𝑁 ) (Δ𝐸Δ𝑁Δ𝑈

)

(4.15)

with 𝐸, 𝑁 and the three axis of a right­handed local topocentric system centered at theobservation point (𝜑, 𝜆, ℎ). The inverse relation of Eq. (4.15) is,

(Δ𝐸Δ𝑁Δ𝑈

) = ( 𝐸 𝑁 )−1 (Δ𝑋Δ𝑌Δ𝑍

) = ( 𝐸 𝑁 )𝑇 (Δ𝑋Δ𝑌Δ𝑍

) (4.16)

where we used the property that the inverse of a rotation matrix is the transpose of the matrix.The vectors 𝐸, 𝑁 and form the three axis of a right­handed local topocentric systemcentered at the observation point (𝜑, 𝜆, ℎ), with z­axis along the normal of the ellipsoid , x­axis 𝐸 orthogonal to the plane of the meridian and positive to the East, and y­axis 𝑁 = ×𝐸in the plane of the meridian completing the topocentric system. The azimuth is counted byconvention from the North and towards the East. For example, in a local topocentric systema point to the North has azimuth 𝛼 of 0∘, and a point to the East +90∘, see Figure 4.5. Theazimuth 𝛼 and zenith angle 𝜁 are defined by,

(Δ𝐸Δ𝑁Δ𝑈

) = 𝑠(sin𝛼 sin 𝜁cos𝛼 sin 𝜁

cos 𝜁) (4.17)

26 4. Spherical and ellipsoidal coordinate systems

with 𝑠 the slant range √Δ𝐸2 + Δ𝑁2 + Δ𝑈2. The inverse relation is,

𝛼 = arctan( Δ𝐸Δ𝑁) = arctan(< 𝐞𝐸 , 𝐬 >< 𝐞𝑁 , 𝐬 >)

𝜁 = arctan( Δ𝑈√Δ𝐸2 + Δ𝑁2

) = arccos(Δ𝑈𝑠 ) = arccos(< , 𝐬 >𝑠 )

𝑠 = √Δ𝐸2 + Δ𝑁2 + Δ𝑈2 = √Δ𝑋2 + Δ𝑌2 + Δ𝑍2 = √< 𝐬, 𝐬 >

(4.18)

with 𝐬 = (Δ𝑋, Δ𝑌, Δ𝑍)𝑇. Compare this to Eq. (3.4), of Chapter 3, where the azimuth and zenithangles were defined in general terms of 𝑋, 𝑌 and 𝑍 coordinates for a topocentric system,whereas in this section the coordinates for the topocentric system have been named 𝐸 (East),𝑁 (North) and 𝑈 (Up) to emphasize the topocentric nature of the system. If the azimuth 𝛼and zenith angle 𝜁 is computed in point A, the origin of the topocentric system, as shownin Figure 4.5, then (Δ𝐸, Δ𝑁, Δ𝑈) in Eqs. (4.15), (4.16), (4.17) and (4.18) can be replacedby (𝐸, 𝑁, 𝑈), and (Δ𝑋, Δ𝑌, Δ𝑍) equals (𝑋 − 𝑋𝐴, 𝑌 − 𝑌𝐴, 𝑍 − 𝑍𝐴), with (𝑋𝐴, 𝑌𝐴, 𝑍𝐴) the 3D globalCartesian coordinates of point A computed by Eq. (4.4).

The order of the coordinates in Eqs. (4.15) and (4.16) is sometimes changed, with the 𝑁(North) coordinates given before the 𝐸 (East) coordinate. This forms a left–handed coordinatessystem and is called a North–East–Up (NEU) system. It follows the common practive forgeographic coordinates of giving the latitude before the longitude.

The coordinates 𝑁 and 𝐸 (Δ𝑁 and Δ𝐸) are sometimes also referred to as Northing andEasting, however, Northing and Easting have been defined before in Eq. (4.7) of Section 4.2,and the two are not exactly the same.

In Section 4.2, differential ellipsoidal coordinates 𝑑𝜑 and 𝑑𝜆), given in radians, whereexpressed in Northing 𝑑𝑁 and Easting 𝑑𝐸 in meters, using the relation of Eq. (4.7)

𝑑𝑁 = ((𝜑) + ℎ) 𝑑𝜑𝑑𝐸 = ((𝜑) + ℎ) cos𝜑 𝑑𝜆

with 𝑑𝑁 the differential in North­South (latitude) direction, with positive direction to the North,and 𝑑𝐸 the differential in East­West (longitude) direction. (𝜑) and (𝜑) are the meridianradius of curvature and radius of curvature in the prime vertical as given by Eq. (4.6) andFigure 4.4. 𝑑𝜑 and 𝑑𝜆 must be given in units of radians, but 𝑑𝑁 and 𝑑𝐸 are in units ofmeters.

The difference with Δ𝑁 and Δ𝐸 is that 𝑑𝑁 and 𝑑𝐸 are curvilinear coordinates, whereas𝑁 and 𝐸 are Cartesian coordinates. For small values of Δ𝑁 and Δ𝐸 we have 𝑑𝑁 ≃ Δ𝑁 and𝑑𝐸 ≃ Δ𝐸, but although 𝑑ℎ ≃ Δ𝑈, the surface 𝑑ℎ = const represents a curved surface,whereas Δ𝑈 = const is a plane tangent to the curved Earth. While we may get away with theapproximation 𝑑𝑁 ≃ Δ𝑁 or 𝑑𝐸 ≃ Δ𝐸, it is a bad idea to mix 𝑈 and ℎ.

The algorithms that involve (Δ𝑋, Δ𝑌, Δ𝑍), (Δ𝑁, Δ𝐸, Δ𝑈), or, the azimuth 𝛼 and zenith angle𝜁 can be used with very large values, provided (Δ𝑁, Δ𝐸, Δ𝑈) are interpreted as coordinatesin a local topocentric left­handed system. The latter comes in very useful for computation ofazimuth and zenith angles of Earth satellites for a point on the surface of the Earth.

The algorithms that involve 𝑑𝜑, 𝑑𝜆 and 𝑑ℎ, or 𝑑𝑁, 𝑑𝐸 and 𝑑ℎ, are only valid for smallvalues or over the surface of the Earth. They are useful mainly for observations are made ateccentric stations, or to transform velocities in the Cartesian system to velocities in the ellip­soidal system, or to propagate error estimates (standard deviations, variances, co­variances)from the Cartesian system into the ellipsoidal system, or vice­versa.

4.5. Practical aspects of using latitude and longitude 27

4.5. Practical aspects of using latitude and longitudeThe ellipsoidal height differs by not more than 100 m from an equipotential surface, or a trueheight coordinate surface, and the ellipsoidal normals agree with the true vertical to within afew seconds of arc. There are no other simple rotational shapes that would match the trueEarth better than an ellipsoid.

The ellipsoidal, geodetic, or geographical, latitude and longitude are therefore the mostcommon representation to describe the position of points on the Earth. And invariably, whenwe use latitude and longitude without any further reference, this is almost always the ellip­soidal, geodetic or geographic latitude and longitude! However, the geodetic latitude 𝜑 shouldnever be confused with the geocentric or spherical latitude 𝜓, or astronomical latitude 𝜙, whichare two different types of coordinates. Also you should not confuse geodetic longitude 𝜆 withastronomical longitude Λ. For the ellipsoidal height ℎ it is a different story. Because the ellip­soid is off from an equipotential surface by up to 100 meters, the ellipsoidal height ℎ is not asuitable coordinate for a height reference system. But, contrary to Cartesian coordinates, withellipsoidal coordinates we can easily separate between ’horizontal’ and ’vertical’ coordinates.And, as you will see in Chapter 8 on height systems, we can replace the ellipsoidal height ℎby the orthometric height 𝐻, with 𝐻 = ℎ−𝑁(𝜑, 𝜆) and 𝑁(𝜑, 𝜆) the so­called geoid height (SeeChapter 7). This leaves us with a couple of options to represent positions

• Use Cartesian coordinates 𝑋, 𝑌 and 𝑍 to represent positions in three dimensions

• Use geographic coordinates and/or height

– Ellipsoidal latitude 𝜑 and longitude 𝜆 to represent positions on the surface of theEarth, with, or, without

– Ellipsoidal height ℎ or orthometric height 𝐻 to represent the vertical dimension

The latitude and longitude can be used with, and, without height information, or viceversa. In case no height information is provided it may be assumed that the positions areon the ellipsoid or another reference surface. The reference surface can be a theoreticalsurface, such as the ellipsoid, or modeled by a digital (terrain) model (with heights givenon a regular grid or as a series of (base) functions).

The mesh formed by the lines of constant latitude and constant longitude forms a graticulethat is linked to the rotation axis of the Earth, as is shown in Figure 4.6. The poles arewhere the axis of rotation of the Earth intersects the reference surface. Meridians are lines ofconstant longitude that run over the reference surface from the North Pole to the South pole.By convention, one of these, the Prime Meridian, which passes through the Royal Observatory,Greenwich, England, is assigned zero degrees longitude. The longitude of other places is givenas the angle East or West from the Prime Meridian, ranging from 0∘ at the Prime Meridian to180∘ Eastward (written 180∘ E or +180∘) and 180∘ Westward (written 180∘ W or −180∘) ofthe Prime Meridian. The plane through the center of the Earth and orthogonal to the rotationaxis intersects the reference surface in a great circle is called the equator. A great circle isthe intersection of a sphere and a plane which passes through the center point of the sphere,otherwise the intersection is called small circle. Planes parallel to the equatorial plane intersectthe surface in circles of constant latitude; these are the parallels. Parallels are small circles.The equator has a latitude of 0∘, the North pole has a latitude of 90∘ North (written 90∘ N or+90∘), and the South pole has a latitude of 90∘ South (written 90∘ S or −90∘). The latitudeof an arbitrary point is the angle between the equatorial plane and the radius to that point.

Degrees of longitude and latitude can be sub­divided into 60 minutes of arc, each of whichis divided into 60 seconds of arc. A longitude or latitude is thus specified in sexagesimal

28 4. Spherical and ellipsoidal coordinate systems

30° W

0°

30° E 60° E

90° E

120° E

45° S

30° S

15° S

0°

15° N

30° N

45° N

60° N

75° N

90° N

25o45’N67o01’E

Figure 4.6: Latitude and longitude grid as seen from outer space (orthographic azimuthal projection). The primemeridian (through Greenwich) and equator are in black with latitude and longitude labels. The meridian andparallel through Karachi, Pakistan, 25∘45′N 67∘01′E, are the dotted lines in red. Meridians are great circles withconstant longitude that run from North to South. Parallels are small circles with constant latitude (the equator isalso a great circle).

notation as 23∘27′30” [EWNS]. The seconds can include a decimal fraction. An alternativerepresentation uses decimal degrees, whereby degrees are expressed as a decimal fraction:23.45833∘ [EWNS]. Another option is to express minutes with a fraction: 23∘27.5′ [EWNS].The [EWNS] 3 suffix can be replaced by a sign: the convention is to use a negative sign forWest and South, and a positive sign for East and North. Further, for calculations decimaldegrees may be converted to radians. Note that the longitude is singular at the Poles andcalculations that are sufficiently accurate for other positions, may be inaccurate at or near thePoles. Also the discontinuity at the ±180∘ meridian must be handled with care in calculations,for example when subtracting or adding two longitudes.

One minute of arc of latitude measured along the meridian corresponds to one nauticalmile (1, 852 m). The nautical mile, which is a non­SI unit, is very popular with navigators inshipping and aviation because of its convenience when working with nautical charts (whichoften have a varying scale): a distance measured with a chart divider can be converted tonautical miles using the chart’s latitude scale. This only works with the latitude scale, butnot the longitude scale, which follows directly from Eq. (4.7) (on account of the term cos𝜑,which result in the meridians converging at the poles), as is shown in Figure 4.7 for the North­Sea area. From Eq. (4.7) it also follows that one degree of arc of latitude measured alongthe meridian is between 110.57 km at the equator and 111.69 km at the poles4. Thus, at

3In Dutch we use the terms O.L. (Oosterlengte) for E, W.L. (Westerlengte) for W, N.B. (Noorderbreedte) for N andZ.B. (Zuiderbreedte) for S.4In surveying and geodesy a circle is not divided in 360∘ but in 400 gon or grad. This has the added advantagethat one gon (grad) measured along the meridian corresponds to 100 km, and one milli­gon (milli­grad) to100 m, a decimilligon (decimilligrad, 10−4 grad) to 10 m and 10−7 gon (grad) corresponds to 1 cm. However,this “decimal” system for angular measurement never gained a big following outside surveying. But, be aware,

4.6. Spherical and ellipsoidal computations [*] 29

10.0° W 7.5° W 5.0° W 2.5° W 0.0° 2.5° E 5.0° E 7.5° E 10.0

° E 50.0° N

52.5° N

55.0° N

57.5° N

60.0° N

0 100 200 300 400 500 km100

0 100 200 300 nm50

Figure 4.7: The latitude and longitude grid over the North­Sea in an equidistant conic projection of uniform scale.One degree of latitude is about 60 nm (Nautical miles), or more precisely 111.2 km at 50∘ and 111.4 km at 60∘latitude. However, one degree of longitude is much shorter, it varies between 71.7 km on the 50∘ parallel and just55.8 km on the 60∘ parallel. This is because the meridians converge to the North.

52∘ latitude, one arc­second (1″) along the meridian corresponds to roughly 30.9 m and onearc­second along the 52∘ parallel to roughly 19.0 m.

Latitude and longitude are angular measures that work well to pin­point a position, but,calculations using the latitude and longitude can be quite involved. For example, the compu­tation of distance, angles and surface area, is far from straightforward and very different fromcomputations using two­dimensional Cartesian coordinates. In general users are left with twooptions: (1) use spherical or ellipsoidal computations, or, (2) first map the latitude and lon­gitude to two­dimensional Cartesian coordinates 𝑥 and 𝑦, and then do all the computationsin the two­dimensional (map) plane. The second option involves a so­called map projection.Computations on the sphere or ellipsoid are discussed in Section 4.6, map projections arediscussed in Chapter 5.

4.6. Spherical and ellipsoidal computations [*]Distances have different meanings. For instance, the distance between an observer in Delftand a satellite orbiting the Earth is the straight line distance computed from the 3D Cartesiancoordinates of both points. If the coordinates of the observer are given in geographical co­ordinates, these are first converted into Cartesian coordinates; something for which also theheight above the ellipsoid is needed (unless the station is assumed to lie on the ellipsoid). Onthe other hand, for the distance between two places on the ellipsoid, say Delft (NL) and SanDiego (CA, USA), the shortest distance over the sphere or ellipsoid is required, and not thestraight line distance.

The equivalent of a straight line in Euclidean geometry for spherical and ellipsoidal geom­

quite often surveying equipment uses gon or grad to measure arcs instead of degrees.

30 4. Spherical and ellipsoidal coordinate systems

140° W 120° W 100° W 80° W 60° W 40° W 20° W 0° 20° E 0°

20° N

40° N

60° N

Figure 4.8: Great circle (blue), rhumb line or loxodrome (red) and straight line (black dashed) distance betweenDelft, NL, 52∘N 4.37∘E and San Diego, CA, USA, 32.8∘N 117.1∘W. The plot on the left uses an orthographic azimuthalprojection, with the Earth as seen from outer space, while the plot on the right uses the Mercator projection. Thegreat circle, rhumb line and straight line distances are 9005, 10077 and 8294 km respectively. The straight line(black dashed) passes through the Earth lower mantle, with a deepest point of 1529 km below the Hudson Strait,Northern Canada, 61.0∘N 71.7∘W. This is also the half­way point for a traveler following the great circle route(blue), which is the shortest route over the Earth surface from Delft to San Diego. The course a traveler is steeringon this route varies between NW (313.5∘) when leaving Delft and SSW (212.1∘) when arriving in San Diego. Arhumb line on the other hand crosses meridians always at the same angle. A traveler following the rhumb lineor loxodrome (red) from Delft to San Diego would have to steer a constant WSW course (257.8∘). Rhumb linesbecome straight lines in a Mercator projection.

etry is the shortest path between points on a sphere or ellipsoid, which is called geodesic (inDutch: de geodetische lijn). On a sphere geodesics are great circles5. This is illustrated inFigure 4.8. Similar geometric concepts are defined in spherical and ellipsoidal geometry as inEuclidean geometry, replacing straight lines by great circles and geodesics. For instance, inspherical geometry angles are defined between great circles, resulting in spherical trigonom­etry.

The solution of many problems in geodesy and navigation, as well as in some branches ofmathematics, involve finding solutions of two main problems:

Direct (first) geodetic problem Given the latitude 𝜑1 and longitude 𝜆1 of point P1, andthe azimuth 𝛼1 and distance 𝑠12 from point P1 to P2, determine the latitude 𝜑2 andlongitude 𝜆2 of point P2, and azimuth 𝛼2 in point P2 to P1.

Inverse (second) geodetic problem Given the latitude 𝜑1 and longitude 𝜆1 of point P1,and latitude 𝜑2 and longitude 𝜆2 of point P2, determine the distance 𝑠12 between pointP1 and P2, azimuth 𝛼1 from P1 to P2, and azimuth 𝛼2 from P2 to P1.

On a sphere the solutions to both problems are (simple) exercises in spherical trigonometry. Onan ellipsoid the computation is much more involved. Work on ellipsoidal solutions was carriedout by for example Legrendre, Bessel, Gauss, Laplace, Helmert and many others after them.The starting point is writing the geodesic as a differential equation relating an elementary

5A great circle is the intersection of a sphere and a plane which passes through the center point of the sphere,otherwise the intersection is called small circle.

4.6. Spherical and ellipsoidal computations [*] 31

segment with azimuth 𝛼 and length 𝑑𝑠 to differential ellipsoidal coordinates (𝑑𝜑, 𝑑𝜆),

𝑑𝜑𝑑𝑠 =

cos𝛼(𝜑)

𝑑𝜆𝑑𝑠 =

sin𝛼(𝜑) cos𝜑

(4.19)

with (𝜑) the meridian radius of curvature and (𝜑) the radius of curvature in the primevertical as given by Eq. (4.6) and Figure 4.4, and with (𝜑) cos𝜑 the radius of the circle oflatitude 𝜑. See also Eq. (4.7) which gives similar relations for Northing 𝑑𝑁 and Easting 𝑑𝐸.These equations hold for any curve. For specific curves the variation of the azimuth 𝑑𝛼 mustbe specified in relation to 𝑑𝑠. For example, for the rumbline, the curve that makes equalangles with the local meridian, 𝑑𝛼/𝑑𝑠 = 0. For the geodesic this relation is

𝑑𝛼𝑑𝑠 = sin𝜑𝑑𝜆𝑑𝑠 =

tan𝜑(𝜑) sin𝛼 (4.20)

Eqs. (4.20) and (4.19) form a complete set of differential equations for the geodesic. Thesedifferential equations can be used to solve the direct and inverse geodetic problems numer­ically. Other solutions involve evaluating integral equations that can be derived from thesedifferential equations. In geodetic applications where 𝑓 is small, the integrals are typicallyevaluated as a series or using iterations. The treatment of this complicated topic goes beyondthe level of this reader.

On a sphere the solution of the direct and inverse geodetic problem can be found usingspherical trigonometry resulting in closed formula. These formula are important for navigation.

Finding the course and distance through spherical trigonometry is a special application ofthe inverse geodetic problem. The inital and final course 𝛼1 and 𝛼2, and distance 𝑠12 alongthe great circle, are

tan𝛼1 =sin 𝜆12

cos𝜙1 tan𝜙2 − sin𝜙1 cos 𝜆12tan𝛼2 =

sin 𝜆12− cos𝜙2 tan𝜙1 + sin𝜙2 cos 𝜆12

cos𝜎12 = sin𝜙1 sin𝜙2 + cos𝜙1 cos𝜙2 cos 𝜆12

(4.21)

with 𝜆12 = 𝜆2 − 𝜆1. 6 The distance is given by 𝑠12 = 𝑅 𝜎12, where 𝜎12 is the central angle(in radians) between the two points and 𝑅 the Earth radius.For practical computations thequadrants of the arctangens are determined by the signs of the numerator and denominatorin the tangent formulas (e.g., using the atan2 function). Using the mean Earth radius yieldsdistances to within 1% of the geodesic distance on the WGS­84 ellipsoid.

Finding way­points, the positions of selected points on the great circle between P1 and P2,through spherical trigonometry is a special application of the direct geodetic problem. Giventhe initial course 𝛼1 and distance 𝑠12 along the great circle, the latitude and longitude of P2

6Please note that in this equation the𝜙 is used for the latitude, but strictly, since this is a computation on the sphere,we should have used the geocentric latitude 𝜓. However, as these formules are often used as an approximationto the more difficult problem on the ellipsoid, you find them often expressed in 𝜙 instead of 𝜓.

32 4. Spherical and ellipsoidal coordinate systems

are found by,

tan𝜙2 =sin𝜙1 cos𝜎12 + cos𝜙1 sin𝜎12 cos𝛼1

√(cos𝜙1 cos𝜎12 − sin𝜙1 sin𝜎12 cos𝛼1)2 + (sin𝜎12 sin𝛼1)2

tan 𝜆12 =sin𝜎12 sin𝛼1

cos𝜙1 cos𝜎12 − sin𝜙1 sin𝜎12 cos𝛼1tan𝛼2 =

sin𝛼1cos𝜎12 cos𝛼1 − tan𝜙1 sin𝜎12

(4.22)

with 𝜎12 = 𝑠12/𝑅 the central angle in radians and 𝑅 the Earth radius, and 𝜆2 = 𝜆1 + 𝜆12.Computations on the sphere, let alone the ellipsoid, are quite complicated. Other tasks

than the direct and inverse geodetic problem, such as the computation of the area on asphere or ellipsoid, which is simple in a 2D Cartesian geometry, require even more complicatedcomputations. Instead a different approach can be taken, which consists of a mapping of thelatitude and longitude (𝜑, 𝜆)𝑖 to grid coordinates (𝑥, 𝑦)𝑖 in a 2D Cartesian geometry, known asmap projection.

4.7. Problems and exercisesQuestion 1 The geographic position coordinates of a geodetic marker in Vlissingen are givenas 51∘26′34.3501″ North, 3∘35′50.3686″ East (in ETRS89). Express the geographic positioncoordinates (latitude and longitude) in decimal degrees.Answer 1 Going from an angle expressed in degrees, minutes and seconds to decimal de­grees, means taking the amount of degrees, adding the number of minutes divided by 60,and adding the number of seconds divided by 3600. This yields 𝜑 = 51.442875∘ North,𝜆 = 3.597325∘ East.

Question 2 The geographic position coordinates of a geodetic marker on Terschelling aregiven as 53.362736∘ North, and 5.219386∘ East (in ETRS89). Express the geographic positioncoordinates (latitude and longitude) in degrees, arcminutes and arcseconds.Answer 2 Going from an angle expressed in decimal degrees to degrees, minutes and sec­onds of arc, means taking the decimal part and multiplying it by 60 and the integer partyields the number of minutes; next taking the original decimal part again, and subtract­ing the integer number of minutes divided by 60, and multiplying this by 3600. This yields𝜑 = 53∘21′45.8496″ North, 𝜆 = 5∘13′9.7896″ East.

Question 3 For the WGS84­ellipsoid, the semi­major axis is given as 𝑎 = 6378137.000 m.And the flattening is 𝑓 = 1/298.257223563. Compute the length of the semi­minor axis 𝑏,and also the eccentricity 𝑒.Answer 3 The eccentricity 𝑒 and semi­minor axis follow from Eq. (4.3), this results in 𝑒 =0.081819191 and 𝑏 = 6356752.314 m, hence the distance from the pole to the Earth’s centeris about 21 km shorter than the distance from the equator to the Earth’s center.

Question 4 The geographic position coordinates of a geodetic marker on Terschelling aregiven as 𝜑 = 53.362736∘ North, 𝜆 = 5.219386∘ East (in ETRS89), and ℎ = 56.098 m. Expressthe position coordinates in Cartesian coordinates. The ellipsoidal parameters of the WGS84ellipsoid can be found in Table 6.1.Answer 4 Converting geographic coordinates into Cartesian coordinates is done throughEq. (4.4), with the expression for the radius of curvature in the prime vertical in Eq. (4.6).At the given latitude, the radius is (𝜑) = 6391928 m. The Cartesian coordinates are𝑋 = 3798580.857 m, 𝑌 = 346993.872 m, 𝑍 = 5094780.835 m.

4.7. Problems and exercises 33

Question 5 The Cartesian coordinates of a location (in the Atlantic Ocean) are given as𝑋 = 6378137.000 m, 𝑌 = 0.000 m, 𝑍 = 0.000 m. Compute, using the WGS84­ellipsoid (seeTable 6.1), the geographic coordinates of this location.Answer 5 The formal computation goes through Eqs. (4.5) and (4.6), and requires an iter­ation, see Eq. (4.8). However, in this special case, as we note that 𝑍 = 0.000 m, we canimmediately conclude that this location lies in the equatorial plane, and latitude 𝜑 = 0∘. Thelongitude follows easily as 𝜆 = 0∘. And eventually the ellipsoidal height ℎ = 0.000 m, as theradius of curvature in the prime vertical equals = 𝑎 = 6378137.000 m, see also Figure 4.4.

Question 6 The position of a GPS receiver on the Delft campus is computed in 2015 usingtwo different processing services: NETPOS and NRCAN. The result from the NETPOS pro­cessing service, given in ETRS89, is 𝜑1 = 51∘59′50.80858” North, 𝜆1 = 4∘22′33.0427” Eastand ℎ1 = 43.5579 m. The result from the NRCAN processing service, given in ITRF2008, is𝜑2 = 51∘59′50.82510” North, 𝜆2 = 4∘22′33.0659” East and ℎ2 = 43.5490 m. After conver­sion to ETRS89 the coordinates from the NRCAN processing are 𝜑3 = 51∘59′50.80910” North,𝜆3 = 4∘22′33.0433” East and ℎ3 = 43.5513 m. Compute the differences in meters betweenthe NETPOS and NRCAN processing, both in ETRS89, and compute the differences in metersbetween the ITRF2008 and ETRS89 solutions for NRCAN.Answer 6 It is clear that the differences are very small, only a fraction of a second of arc.The difference between the NETPOS and NRCAN solution, both in ETRS89, is Δ𝜑 = 𝜑3−𝜑1 =0.00052” , Δ𝜆 = 𝜆3−𝜆1 = 0.0006” and Δℎ = ℎ3−ℎ1 = −0.0266 m. To convert the differencesinto units of meters Eq. (4.7) is used. At 52∘ latitude we have

Δ𝑁[m] = 𝜋180 ∗ 3600 ∗ 6391000 ∗ Δ𝜑[”] ≃ 31.0 ∗ Δ𝜑[”]

Δ𝐸[m] = 𝜋180 ∗ 3600 ∗ 6376000 ∗ cos(𝜑) ∗ Δ𝜆[”] ≃ 19.0 ∗ Δ𝜆[”]

whereby we obtained (𝜑) ≃ 6391 km and (𝜑) ≃ 6376 km from Figure 4.4 or Eq. (4.6).Note that 𝑅 = 6371 km instead of (𝜑) and (𝜑) would have given a more or less similarresult. The difference between the NETPOS and NRCAN solution is thus Δ𝑁 = 31∗0.00052” =0.0161 m , Δ𝐸 = 19∗0.0006” = 0.0114 m and Δℎ = −0.0266 m. The differences between thetwo solutions are in the order of centimeters.

The difference between the ETRS89 and ITRF2008 solution is Δ𝜑 = 𝜑2 − 𝜑3 = 0.01600”,Δ𝜆 = 𝜆2 − 𝜆3 = 0.0226” and Δℎ = ℎ2 − ℎ3 = −0.0177 m. To convert the differences intounits of meters again Eq. (4.7) is used, which results in Δ𝑁 = 31 ∗ 0.016” = 0.496 m, Δ𝐸 =19.088 ∗ 0.0226” = 0.429 m and Δℎ = −0.0177 m.

Please note that the horizontal differences between the ITRF2008 and ETRS89 solutionsof NRCAN, at decimeter level, are much larger than the differences between NRCAN andNETPOS solutions in the same reference frame. This is due to ETRS89 moving along with theEuropean plate, with station velocities in Europe close to zero, whereas in ITRF2008 Delft ismoving yearly 2.3 cm to the North­East. Over a period of 26 years (the epoch of observation,2015, minus 1989, the year ETRS89 and ITRF2008 coincided), this corresponds to about0.60 m. See also Chapter 9 for more information. It also shows the importance of datumtransformations, which we used to convert ITRF2008 coordinates to ETRS89, and is the topicof Chapter 6.

5Map projections

Geographic latitude and longitude are convenient for expressing positions on the Earth, butcomputations on the sphere, let alone the ellipsoid, are quite complicated as we have seen inthe previous Chapter. Instead a different approach can be taken, which consists of a mappingof the latitude and longitude (𝜑, 𝜆)𝑖 to grid coordinates (𝑥, 𝑦)𝑖 in a 2D Cartesian geometry.This is known as a map projection. From then on simple 2D Eucledian metric can be used.

5.1. IntroductionMap projections are used in both cartography and geodesy. The output of a map projectionin cartography is usually a small scale map, on paper, or in a digital format. The requiredaccuracy of the mapping is low and a sphere may be safely used as the surface to be mapped.In cartography it is more about appearance and visual information than accuracy of the coor­dinates.

In geodesy a map projection is more a mathematical device that transfers the set of geo­graphical coordinates (𝜑, 𝜆) into a set of planar coordinates (𝑦, 𝑥) without loss of information.The relation can therefore also be inverted (i.e. undone). It implies that an ellipsoid shouldbe used as the surface to be mapped. This also applies for medium and large scale maps, andcoordinates that are held digitally in a Geographic Information System (GIS) or other informa­tion system. In this reader a map projection is defined as the mathematical transformation

𝑦 = 𝑓(𝜑, 𝜆, ℎ)𝑥 = 𝑔(𝜑, 𝜆, ℎ) (5.1)

whereby ℎ is implicitly given as zero (ℎ = 0), meaning points are first projected on the ellipsoid.The coordinates (𝑥, 𝑦) are called map or grid coordinates. The grid coordinates are oftenreferred to as Northing (𝑦) and Easting (𝑥).

Many different map projections are in use all over the world for different applications andfor good reasons. However, having many different types of map projections and grid coor­dinates, may sometimes also result in confusion about what coordinates are actually used orgiven. Some software packages may support many of these map projections, but it is virtuallyimpossible to support them all. Other softwares are specifically written for one specializedmap projection, and give incorrect results when using coordinates from a different type ofprojection.

35

36 5. Map projections

Cylindrical Conic Azimuthal

Figure 5.1: Cylindrical, Conic and Azimuthal map projection types (Source: Wikimedia Commons).

5.2. Map projection types and propertiesMap projections can be grouped into four groups depending on the nature of the projectionsurface, see Figure 5.1,

Cylindrical map projections The plane of projection is a cylinder wrapped around theEarth. Cylindrical projections are easily recognized for its shape: maps are rectangularand meridians and parallels are straight lines crossing at right angles. A well knownprojection is the Mercator projection. Figure 4.8 (right part) of the previous chapter is aMercator projection.

Conic map projections The plane of projection is a cone wrapped around the Earth. Par­allels become arcs of concentric circles. Meridians, on the other hand, converge tothe North or South. Often used for regions of large east­west extent. An example isFigure 4.7 of the previous chapter.

Azimuthal map projections The plane of projection is a plane tangent to the Earth. Twowell known examples are the stereographic projection, which is used for instance by theDutch RD system, see Chapter 10, and the orthographic azimuthal projection used inFigure 4.6 of the previous chapter.

Miscellaneous projections Mostly used for cartographic purposes.

Any projection can be applied in the normal, oblique and transverse position of the cylinder,cone or tangent plane, as shown in Figure 5.2 for a cylinder. In the normal case the axis ofprojection, the axis of the cylinder and cone, or normal to the plane, coincides with the minoraxis of the ellipsoid.

An example of a cylindrical map projection is the Mercator projection, with the equator asthe line of contact of the cylinder, see Section 5.4.2 and Figure 4.8. In the transverse casethe axis of projection is in the equatorial plane (orthogonal to the minor axis), for example,in the Universal Transverse Mercator (UTM) projection small strips are mapped on a cylinderwrapped around the poles and with a specific meridian as line of contact. In the oblique casethe axis of projection does not coincide with the semi­minor axis or equatorial plane.

Map projections also differ in the point of perspective that is used. For instance, the pointof perspective for the azimuthal stereographic projection is a point on the Earth oppositeto the tangent plane, as is depicted in Figure 5.1 on the right. On the other hand, for theorthographic azimuthal projection, which was used for Figure 4.6 and in Figure 4.8 (left part),

5.2. Map projection types and properties 37

the point of perspective is at infinite distance. The orthographic azimuthal projection depictsa hemisphere of the globe as it appears from outer space, which results in shapes and areasdistorted particularly near the edges.

normal transverse oblique

λ φ

a

P

x

λ=0

y

N

S

P'

centralmeridian

P

y

N

S

Px

N

S

y

x

centralmeridian

centralgreat circle

Figure 5.2: Normal, transverse and oblique projection for a cylinder (Source: Wikimedia Commons).

Some distortion in the geometrical elements, distance, angles and area, is inevitable inmap projections. In this respect map projections are divided into

Conformal projections Preserves the angle of intersection of any two curves.

Equal Area (equivalent) projections Preserves the area or scale.

Equi Distance (conventional) projections Preserves distances.

Map projections may have one or two of these properties, but never all three together. Ingeodesy conformal mappings are preferred. A conformal mapping may be considered a simi­larity transformation (see Section 2.2) in an infinitesimally small region. A conformal mappingdiffers only from a similarity transformation in the plane in that its scale is not constant butvarying over the area to be mapped. For cartographic purposes, e.g. employing geostatistics,equal area mappings may be better suited.

In some projections an intermediate sphere is introduced. These are called double projec­tions; the first step is a conformal mapping onto a sphere, the second step is the subsequentprojection from the sphere onto a plane. This is also the basis for the Dutch map projection:the first step is a conformal Gauss projection from the Bessel ellipsoid on the sphere, thesecond step a stereographic projection onto a plane tangential to the ellipsoid with the centerat Amersfoort.

In order to specify a map projection the following information is required

• Name of the map projection or EPSG dataset coordinate operation method code (seeSection 6.4)

• Latitude of natural origin or standard parallel (𝜑0) for cylindrical and azimuthal projec­tions, or, the latitude of first standard parallel (𝜑1) and second standard parallel (𝜑2) forconic projections

• Longitude of natural origin (the central meridian) (𝜆0)

• Optional scale factor at natural origin (on the central meridian)

38 5. Map projections

• False Easting and Northing

The false Easting and Northing are used to offset the planar coordinates (𝑥, 𝑦) in order toprevent negative values.

5.3. Practical aspects of map projectionsWorking with planar grid coordinates to compute distances, angles and areas is much moreconvenient than using geographical coordinates. However, one should be aware that in themap projection small distortions are introduced. For example, an azimuth computed from gridcoordinates may not be referring to true North because of meridian convergence in azimuthaland conic projections. Meridian convergence is defined as the angle meridians make withrespect to the grid y­axis. Also, sometimes corrections need to made for distances and surfaceareas. These corrections are usually quite small and well known. If they become too large itmay be necessary to reduce the area of the projection, e.g. by defining different zones, eachwith a different natural origin or central meridian (or parallel). This approach is for instanceused by the popular Universal Transverse Mercator (UTM) projection, see Section 5.4.5, whichuses between 80∘S and 84∘N latitude 60 zones, each of 6∘ width in longitude, centered arounda central meridian. However, the Netherlands falls in two zones, 31N and 32N, which isnot very convenient and may explain why the UTM projection is not used very often in theNetherlands except off­shore on the North Sea. UTM has also been the projection of choicefor the European Datum 1950 (ED50).

Map projections are usually equations that provide a relationship between latitude and lon­gitude on the one hand, and planar grid coordinates on the other hand. However, sometimesthe transformation to planar coordinates, and vice versa, may be supplemented by tabulatedvalues in the form of a correction grid to account for local distortions in the planar grid coor­dinates. This is often the case when the planar grid has been based on first order geodeticnetworks established in the 19th and early 20th century using triangulations, pre­dating themore accurate satellite based techniques in use today. These older measurements, althoughquite an achievement in their time, typically resulted in long wavelength (> 30 km) distor­tions in the first order networks, which were the basis for all other (secondary and lower order)measurements, and are therefore present in all planar grid coordinates. In order for satellitedata, which are not related to the first order networks, to be transformed into planar gridcoordinates and to used together with already existing data, many national mapping agenciesdecided to adopt a conventional correction grid to their planar coordinates. So, if the planarcoordinates are converted into latitude and longitude (to be used together with other satellitedata), the correction grid corrects for distortions in the planar grid coordinates. If, on theother hand, latitude and longitude is converted to grid coordinates, (conventional) distortionsare re­introduced so that the satellite data, expressed in grid coordinates, matches existingdatasets.

5.4. Cylindrical Map projection examplesIn this section several examples of cylindrical projections are presented. Cylindrical projectionshave been chosen because the mathematics are less complicated than those of other mapprojections, and thus serve well to illustrate some principles of map projections. Some of thecylindrical projections that are discussed are only for illustration, but others, like the Mercator,Web Mercator and UTM projections, are used (almost) on an every day basis.

With the cylindrical projection the Earth’s surface is projected onto a cylinder tangent tothe equator, as shown in the left part of Figure 5.3. The map projection turns (spherical)

5.4. Cylindrical Map projection examples 39

Figure 5.3: For the cylindrical projection, the mapping plane is wrapped around the Earth like a cylinder (left),longitude 𝜆 turns into map­coordinate 𝑥 (middle), and latitude 𝜑 turns into map­coordinate 𝑦 (right).

coordinates (𝜑, 𝜆) 1 of points on the Earth’s surface into map or grid coordinates (𝑥, 𝑦). Themap origin (𝑥 = 0, 𝑦 = 0) is at the intersection of the equator and the Greenwich meridan(𝜑 = 0, 𝜆 = 0). In the sequel latitude 𝜑 and longitude 𝜆 are expressed in radians. The Earth’ssurface is approximated by a sphere with radius 𝑅. The middle part of Figure 5.3 shows a topview of the equatorial plane. The distance from (𝜑 = 0, 𝜆 = 0) along the equator to an objectat longitude 𝜆 equals 𝑅𝜆, hence we simply have: 𝑥 = 𝑅𝜆 for all normal cylindrical projections.This is a property of all normal cylindrical projections: points on the meridian have a constant𝑥 value.

The function 𝑦 = 𝑦(𝜑) to project latitude 𝜑 onto 𝑦 values is still open, it can be anyone from an unlimited number of functions. In Figure 5.3, on the right, one such function isillustrated: the central cylindrical projection.

5.4.1. Central cylindrical projectionIn case of the central cylindrical projection points on the Earth are projected, from the originat the middle of the Earth, onto a cylinder tangential to the Earth at the equator. The rightpart of Figure 5.3 shows a meridianal cross section of the Earth at longitude 𝜆. The objectpoint, projected onto the cylinder, has a distance 𝑅 tan𝜑 from the equator, hence we have:𝑦 = 𝑅 tan𝜑. The map­projection equations for the central cylindrical projection are thus

𝑥 = 𝜇𝑅(𝜆 − 𝜆0)𝑦 = 𝜇𝑅 tan𝜑 (5.2)

with 𝜇 a scaling factor and 𝜆0 the central meridial (e.g. Greenwhich meridian with 𝜆0 = 0),To represent coordinates on an actual paper map, or graphical display, a very small scalingfactor 𝜇 is applied, e.g. to obtain a paper map with map­scale 1 ∶ 10000000 you would select𝜇 = 1

10000000 rather than 1.The true scale on the Equator is unity for 𝜇 = 1. Everywhere else the linear scale is

stretched by a factor of 1/ cos𝜑 = sec𝜑 in the 𝑥­axis direction, and 1/ tan𝜑 = cot𝜑 in the 𝑦­axis direction. The central cylindrical projection is neither conformal or equal area. Distortion

1The notation used here is the one for geographical coordinates. The distortions that are inherent to this projectionmake that the use of spherical or geographic coordinates doesn’t matter for a graphical representation. However,formallly, geographic coordinates should first be projected onto spherical coordinates, before the map projectionis applied.

40 5. Map projections

increases so rapidly away from the equator, see Figure 5.4, that the central cylindrical isseldomly used for practical maps. Its vertical, latitudinal, stretching is even greater than thatof the Mercator projection, which we discuss next.

5.4.2. Mercator projectionThe Mercator projection is a cylindrical map projection, presented by the Flemish geographerand cartographer Gerardus Mercator, in 1569. It became the standard map projection fornautical navigation, as a line of constant course, known as rhumb line, see the red­line inFigure 4.8, is shown as a straight line, that conserves the angle with the meridians.

As in all cylindrical projections, parallels and meridians are straight and perpendicular toeach other. The Mercator map­projection is a conformal map projection, meaning that anglebetween any two straight lines or curves is preserved. To this end the East­West stretchingof the map (to ‘undo’ the meridian­convergence), which increases as distance away from theequator increases, is accompanied by a corresponding North­South stretching. The distancebetween the parallels gets larger and larger, the further one gets away from the equator, like inany cylindrical projection, but the amount by which is chosen carefully as to preserve angles.

As the radius of a parallel, or circle of latitude, is 𝑅 cos𝜑, the corresponding parallel on themap, a line with with a constant 𝑦 coordinates has been stretched by a factor of 1/ cos𝜑 =sec𝜑 in the 𝑥­coordinate direction. To preserve angles the same amount of stretching needs tobe applied in the 𝑦­coordinate direction. This implies that the derivative of the map­coordinatesfunction 𝑦(𝜑) must be 𝑦′(𝜑) = 𝑅 sec𝜑. 2 Integrating this equation gives

𝑦(𝜑) = 𝑅 ln [tan(𝜋4 +𝜑2 )] (5.3)

This function is illustrated in Figure 5.4. The map projection formulae for a basic normalMercator projection are thus

𝑥 = 𝜇𝑅(𝜆 − 𝜆0)𝑦 = 𝜇𝑅 ln [tan(𝜋4 +

𝜑2 )]

(5.4)

with 𝜇 a scaling factor and 𝜆0 the central meridian. The true scale on the Equator is unityfor 𝜇 = 1. Everywhere else the linear scale is stretched by a factor of 1/ cos𝜑 = sec𝜑.This distorts the size of geographical objects far from the equator; objects like Greenlandand Antartica appear to be much larger than they in reality are, see Figure 4.8, and alsoFigure 5.4. The Mercator projection is conformal, it preserves angles, but it is definitely not anequal area projection. By choosing a value of 𝜇 slightly smaller than one (effectively decreasingthe radius of the cylinder) we can create a Mercator projection with the unity scale for twoparallels, but this does not solve the problem of distortions. At higher latitude the Mercatorprojection becomes unusable, and even becomes singular at the poles (the North and Southpole become lines at 𝑦 = ∞).

In the previous equations it was assumed that the Earth was modelled by a sphere, or,more precisely, we should have used sperical coordinates (𝜆, 𝜓) instead of geographical coor­dinates (𝜆, 𝜑). To use geographic coordinates instead of sperical coordinates is only a minorapproximation for global small scale maps.

When the Earth is modelled by an ellipsoid, with (𝜆, 𝜑) is the geographic longitude andlatitude, the Mercator projection must be modified to remain conformal. The map projection

2The derivative for the central cylindrical projection is 𝑦′(𝜑) = 𝑅 sec2𝜑.

5.4. Cylindrical Map projection examples 41

0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

[deg]

y(ϕ

)[-

]

Centralucylindrical

0

20

40

60

0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

[deg]

Mercator

0

20

40

60

80

0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

[deg]

Equirectangular

0

20

40

60

80

Figure 5.4: Mapping function 𝑦 = 𝑦(𝜑) for the Central cylindrical, Mercator and Equirectangular (Plate Carrée)projections. The function 𝑦(𝜑) with, 𝑅 = 1, is the black line. The x­axis is 𝜑 in degrees, the y­axis on the left ofeach plot gives the map­coordinate 𝑦 = 𝑦(𝜑), the y­axis on the right of each plot gives the latitude (in degrees)that corresponds to 𝑦(𝜑). The blue lines are coast lines for part of the Earth plotted with the function 𝑦(𝜑) onthe y­axis, with longitude (in degrees) on the x­axis.

formula in case of the ellipsoidal model are

𝑥 = 𝜇𝑅(𝜆 − 𝜆0)

𝑦 = 𝜇𝑅 ln [tan(𝜋4 +𝜑2 ) (

1−𝑒 sin𝜑1+𝑒 sin𝜑)

𝑒2 ] (5.5)

with 𝑒 the excentricity of the ellipsoid.

5.4.3. Plate carrée and equirectangular projectionsA simple longitude­latitude presentation is obtained when the 𝑥­ and 𝑦­coordinates are scaledby 𝑅 in the same way. This is called the plate carrée projection. The map­projection equationsfor this simple cylindrical map­projection are

𝑥 = 𝑅(𝜆 − 𝜆0)𝑦 = 𝑅𝜑 (5.6)

The parallels and meridians are being equidistant in the map and form a square grid3, as canbe seen in Figure 5.4. The scale in the latitude (N­S) direction is uniform, at least for a sphericalEarth. However, the scale for the longitude (E­W) direction is not uniform and decreases withthe latitude. The plate carrée projection is a special case of the equirectangular projection.

The map projection equations for the equirectangular projection, with standard parallels

3With the default Mercator projection the parallels get further and further apart the more to the North (South) yougo, and with the central cylindrical projection this will be even more the case.

42 5. Map projections

at 𝜑1 North and South of the equator, are

𝑥 = 𝑅(𝜆 − 𝜆0) cos𝜑1𝑦 = 𝑅(𝜑 − 𝜑1)

(5.7)

The projection maps meridians to vertical straight lines of constant spacing, and circles oflatitude to horizontal straight lines of constant spacing, to form a rectangular grid. The scaleof the projection is true at both standard parallels 𝜑1. The projection is neither equal area norconformal.

Because of the distortions introduced by this projection it has little use in navigation orcadastral mapping. However, it is an easy to use projection for mapping small areas, and itdoes a much better job than simply plotting longitude and latitude values in an xy­plot, whatthe Plate Carré projection basically does.

5.4.4. Web MercatorThe Web Mercator projection is a variant of the Mercator projection that is used by manyWeb mapping applications, including Google Maps, Bing Maps, OpenStreetMap and others.Its official EPSG identifier is EPSG:3857. It uses the same spherical formulas of Eq. 5.4 asthe standard Mercator, however, the Web Mercator uses the spherical formulas with the geo­graphical coordinates (𝜆, 𝜑) in the WGS 84 ellipsoidal datum. The discrepancy is imperceptibleat the global scale, but causes maps of local areas to deviate slightly from true ellipsoidal Mer­cator maps. This discrepancy also causes the projection to be slightly non­conformal. Forthese reasons, several agency have declared this map projection to be unacceptable for anyofficial use.

5.4.5. Universal Transverse Mercator (UTM)The normal Mercator projection works quite well in a small band around the Equator, butperforms very poorly at higher latitudes. Switching from a normal projection, to a transverseprojection, as in Figure 5.2, results in a projection that works quite well in a small band aroundthe central meridian. This approach is used by the popular Universal Transverse Mercator(UTM) projection for latitudes between 80∘S and 84∘N. The UTM projection uses 60 zones,each of 6∘ width in longitude (up to 668 km), centered around a central meridian. Each zoneis numbered. For instance, the Netherlands falls in two zones, 31N and 32N. Zone 31N coverslongitude 0∘ to 6∘E, zone 32N covers longitude 6∘E to 12∘E.

The scale factor along the central meridian is not 1, but 0.9996, so that the inevitabledistortion is spread more uniformly over the zone. The amount of distortion is less than1/1000 .

In each zone the scale factor of the central meridian reduces the diameter of the transversecylinder to produce a secant projection with two standard lines, or lines of true scale, about180 km on each side of, and about parallel to, the central meridian (Arc cos 0.9996 = 1.62°at the Equator). The scale is less than unity inside the standard lines and greater than unityoutside them, but the overall distortion is minimized

The polar regions South of 80∘S and North of 84∘N are excluded.

5.5. Problems and exercisesQuestion 1We do have a geographic database available, with position coordinates in a three­dimensional Cartesian Earth Centered, Earth Fixed (ECEF) reference system. We would like tocreate a map of the Northern hemisphere, using an orthographic azimuthal projection (withthe mapping plane being parallel with the equatorial plane, and lying/touching the North pole).

5.5. Problems and exercises 43

Set up the 3­by­3 projection matrix to perform the mapping operation on the three dimensionalcoordinates in the database.

Figure 5.5: Orthographic azimuthal map projection. The point of tangency of the mapping plane is the North Pole.

Answer 1 The mapping plane is 𝑍 = 𝑏, with 𝑏 the semi­minor axis of the ellipsoid (or theradius of the sphere). Next, orthographic means that the projection lines are all perpendicularto the mapping plane, and in this case parallel to the Z­axis. Hence the projection matrix is

𝑃 = (1 0 00 1 00 0 0

),

and the Z coordinate is eventually to be translated to 𝑍 = 𝑏. So mapping, or grid coordinates(𝑥, 𝑦) become 𝑥 = 𝑋 and 𝑦 = 𝑌. This is shown in Figure 5.5. Eventually you may want toapply a scale factor 𝜇, so that 𝑥 = 𝜇𝑋 and 𝑦 = 𝜇𝑌.

6Datum transformations and

coordinate conversions

Geospatial projects often involve spatial coordinates from different sources, each using theirown coordinates representation and reference system. In order to establish the correct spatialrelationships, first the coordinates have to be transformed into the same reference system andrepresentation. Transformations between reference systems are also called geodetic datumtransformations. In this chapter we discuss geodetic datum transformations and coordinateconversions.

6.1. Geodetic datumIn the previous chapters several types of spatial coordinate systems and representations havebeen introduced, such as Cartesian coordinates, geographic coordinates and grid (map) co­ordinates, including operations that can be performed on them. But, somehow, spatial coor­dinates need to be linked to the Earth, the so–called geodetic datum. For instance, take theexample of Cartesian and ellipsoidal coordinates first. For 3D Cartesian coordinates we needto define 7 parameters: three for the origin, three for the orientation of the axis, and onefor scale (see Section 3.4). For ellipsoidal coordinates, i.e. geographic latitude, longitude andellipsoidal height, we need to define

• Shape of the ellipsoid: length of semi­major axis and flattening (or length of semi­minoraxis or inverse flattening) of the chosen ellipsoid (2 parameters)

• Position of the ellipsoid with respect to the Earth: origin, orientation and scale of theellipsoid (7 parameters)

When countries developed their national coordinate systems at the end of the 19th and begin­ning of the 20th century, each country choose an ellipsoid of revolution that best fitted theircountry based on astronomical observations. This resulted not only in different choices for theshape of the ellipsoid, but also in different positions of the ellipsoid on the Earth. Table 6.1gives the parameters for three commonly used ellipsoids in the Netherlands. Because of thelimited accuracy of astronomical observations at the time, the position of the ellipsoids alsodiffered.

Therefore, each spatial coordinate systems has a geodetic datum. The geodetic datum,or datum, specifies how a coordinate system is linked to the Earth: it consists of parametersthat describe how to define the origin of the coordinate axis, how to orient the axis, and how

45

46 6. Datum transformations and coordinate conversions

Ellipsoid 𝑎 [m] 1/𝑓 [­] 𝐺𝑀 [𝑚3/𝑠2]Bessel (1841) 6 377 397.155 299.152 812 8GRS80 6 378 137 298.257 222 100... 3 986 005 108WGS84 6 378 137 298.257 223 563 3 986 004.418 108

Table 6.1: Common ellipsoids, with semi­major axis 𝑎, inverse flattening 1/𝑓, and if available, associated valuefor 𝐺𝑀. The full list of ellipsoids is much longer. The very small difference in the flattening between WGS84 andGRS80 results in very tiny differences of at most 0.105 mm and can be neglected for all practical purposes.

scale is defined. However, this can only be done in a relative sense, using a geodetic datumtransformation to link one reference system to the other. The parameters that are involved,usually origin, orientation and scale changes, are the so­called geodetic datum transformationparameters.

6.2. Coordinate operationsIt is common for spatial reference systems to use different geodetic datums, reference el­lipsoids and map projections. Therefore, coordinate transformations between two differentsystems not only involve a 7­parameter similarity transformation, the datum transformation,but often also a change in the reference ellipsoid, type of map projection and projection pa­rameters. Two types of coordinate operations have to be distinguished

datum transformation This changes the datum of the reference system, i.e. how the co­ordinate axis are defined and how the coordinate system is linked to the Earth. Datumtransformations typically involve a 7­parameter similarity transformation between Carte­sian 3D coordinates.

coordinate conversions These are conversions from Cartesian into geographical coordi­nates, geographical coordinates into grid (map) coordinates, geocentric Cartesian intotopocentric, geographic into topocentric, etc., and vice versa. These are operations thatoperate on coordinates from the same datum. In general, one type of coordinates canbe converted into another, without introducing errors or loss of information, as long asno change of datum is involved.

A diagram showing the relations between datum transformations and coordinate conversionsis presented in Figure 6.1.

Datum transformations are transformations between coordinates of two different referencesystems. Usually this is a 7­parameter similarity transformation between Cartesian coordinatesof both systems, as shown in Figure 6.1, but if a dynamic Earth is considered with movingtectonic plates and stations, the similarity transformation can be time dependent (with 14instead of 7 parameters). Affine or polynomial transformations between geographic or gridcoordinates of both systems are also possible, but not shown in Figure 6.1. These affine orpolynomial transformations are mostly course approximations.

Coordinate conversions depend only on the chosen parameters for the reference ellipsoid,such as the semi­major axis 𝑎 and flattening 𝑓, and the chosen map projection and projec­tion parameters. Once these are selected and remain unchanged coordinate conversions areunambiguous and without loss of precision.

The conversion from 3D coordinates to 2D grid (map) or geographic coordinates in Fig­ure 6.1 is very straightforward: this is accomplished by simply dropping the height coordinate.The reverse, from 2D to 3D, is indeterminate. This is an issue when 2D coordinates (geo­graphic or grid coordinates) have to be transformed into another datum or reference system,

6.2. Coordinate operations 47

𝑥, 𝑦 𝜑, 𝜆

𝑋, 𝑌, 𝑍

𝐻 ℎ

𝑥, 𝑦 𝜑, 𝜆

𝑋, 𝑌, 𝑍

𝐻 ℎ

map projection

map projection

geoid

geoid

geographic Cartesian

geographic Cartesian

7p

similarity

transfo

rmatio

n

Coordinate conversion

Datu

m tran

sform

ation

A

B

Figure 6.1: Coordinate conversions and datum transformations. Horizontal operations represent coordinate con­versions. The vertical operations are datum transformations from system A to B. Not shown in this diagram arepolynomial transformations (approximations) directly between map coordinates or geographic coordinates of thetwo systems.

as this involved 3D Cartesian coordinates. However, in practice this issue is resolved easily bycreating an artificial ellipsoidal height ℎ, for instance by setting the ellipsoidal height ℎ = 0.The resulting height in the new system will of course be meaningless, and has to be dropped,but as long as the chosen ellipsoidal height is within a few km of the actual height the errorinduced in the horizontal positioning will be small.

A big difference between datum transformations and coordinate conversions is that theparameters for the datum transformation are often empirically determined and thus subjectto measurement errors, whereas coordinate conversions are fully deterministic. More specific,three possibilities need to be distinguished for the datum transformation parameters

1. The first possibility is that the datum transformation parameters are conventional. Thismeans they are chosen and therefore not stochastic. The datum transformation is thenjust some sort of coordinate conversion (which are also not stochastic).

2. The second possibility is that the datum transformation parameters are given, but havebeen derived by a third party through measurements. What often happens is that thisthird party does new measurements and updates the transformation parameters occa­sionally or at regular intervals. This is also related to the concepts of reference systemand reference frames. Reference frames are considered (different) realizations of thesame reference system, with different coordinates assigned to the points in the referenceframe, and often with different realizations of the transformation parameters. The sta­tion coordinates and transformation parameters are stochastic, so new measurements,mean new estimates that are different from the previous estimates.

3. The third possibility is that there is no third party that has determined the transformation

48 6. Datum transformations and coordinate conversions

parameters, and you as user, have to estimate them using at least three common pointsin both systems. In this case you will need coordinates from the other reference system.Keep in mind that the coordinates from the external reference system all come from thesame realization, or, reference frame.

6.3. A brief history of geodetic datumsMany different datums and reference ellipsoids have been used in the history of geodesy. Atthe end of the 19th and beginning of the 20th century many countries developed their ownnational coordinate system, choosing an ellipsoid of revolution that best fitted the area ofinterest. In this pre­satellite era this meant doing astronomical observations to determine theorigin and orientation of the ellipsoid. This resulted in many different ellipsoids and datums.In the 1950’s USA initiated work on ED50 (European Datum 1950) which had as goal to linkthe various European datums and create a European reference system primarily for NATO ap­plications. ED50 became also popular for off­shore work and to define the European borders.The satellite era saw the development of a number of world­wide reference systems, suchas WGS60 and WGS72 which were based on Transit/Doppler measurements, with the mostrecent version WGS84 based on GPS in 1987. Later the International Terrestrial ReferenceFrame (ITRF) and the European Terrestrial Reference System ETRS89 were established whichare more accurate than WGS84. These global reference frames also made it possible for thefirst time to determine accurate datum transformation parameters for the national referenceframes that were established in the 19th and early 20th century.

With the advent of GPS and other space geodetic techniques the newer reference ellipsoidsand datums are all very well aligned to the center of mass and rotation axis of the Earth.These geocentric reference ellipsoids are usually within 100m of the geoid world­wide. In pre­satellite days the reference ellipsoids were devised to give a good fit to the geoid only over thelimited area of a survey, and it is therefore no surprise that there are significant differencesin shape and orientation between the older and newer ellipsoids, resulting in large datumtransformation parameters for the old systems. This also means that there are significantdifferences between latitude and longitude defined on one of the older legacy ellipsoids withrespect to the satellite based datums. Confusing the datums of the latitude and longitude mayresult in significant positioning errors and could result in very hazardous situations.

It is therefore very important with coordinates (does not matter whether they are Cartesian,geographic or grid coordinates) to always specify the reference system and reference framethey belong to. Also, for measurements on a dynamic Earth, it is important to document themeasurement epoch. The reference system and reference frame of the coordinates, and themeasurement epoch, are very important meta data for coordinates which should never beomitted. Failure to record or provide this important meta data will almost always result inconfusion and may result in unnecessary costs.

6.4. EPSG dataset and coordinate conversion softwareThe International Association of Oil & Gas Producers (OGP) maintains a geodetic parameterdataset of common coordinate conversions, datum transformations and map projections. Thisis known as the EPSG dataset (EPSG stands for European Petroleum Survey Group), wherebyeach coordinate operation or transformation is identified by a unique number. In the EPSGDataset codes are assigned to coordinate reference systems, coordinate transformations, andtheir component entities (datums, projections, etc.). Within each entity type, every record hasa unique code (http://www.epsg.org). The EPSG website also provides the equations for

6.5. Problems and exercises 49

the various mappings that have been stored in the EPSG database1. The EPSG database,although extremely useful, has no official status, and sometimes contains only approximateparameters. For high precision applications, that require an accuracy better than 1 meter, theuser should be very careful. For instance, the EPSG code for the Dutch RD coordinate systemis EPSG:28992 (Amersfoort / RD New – Netherlands) https://epsg.io/28992, but theaccuracy is just a few decimeters. This is fine for visualizations on a map or GIS system,but it by no means a substitute for the official coordinate transformation that is discussed inChapter 10.

Software for map projections, coordinate conversions and datum transformations is pro­vided for instance by the open source PROJ package (https://proj.org/) used by severalGeographic Information System (GIS) packages (e.g. the open source QGIS package). PROJstarted purely as a cartography application, but over the years support for datum shifts andmore precise coordinate transformations were added to PROJ. In their own words: “TodayPROJ supports more than a hundred different map projections and can transform coordinatesbetween datums using all but the most obscure geodetic techniques”.

PROJ includes command line applications for conversion of coordinates from text files oruser input, and an application programming interface. Coordinate transformations are definedby string that holds the parameters of a given coordinate transformation, e.g. the examplestring +proj=merc +lat_ts=56.5 +ellps=GRS80 specifies a Mercator projection withthe latitude of true scale at 56.5∘N on the GRS80 ellipsoid. The command proj +proj ...converts the geographic (geodetic) coordinates, read from standard input, to map coordinates.The cs2cs command line utility is used to transform from one coordinate reference system toanother, using two +proj strings to specify the source and destination system. If you knowthe EPSG identifiers these can be used to specify the source and destination.

PROJ supports the Dutch RD coordinate system, see Chapter 10, through EPSG codeEPSG:28992 for the approximate transformation, but recent versions of PROJ can also beused to implement the RDNAPTRANS™2018 procedure using a daisy chain of +proj strings.

6.5. Problems and exercisesQuestion 1 Compute the difference in semi­major axis length between the WGS84 ellipsoidand the Bessel­1841 ellipsoid.Answer 1 The length of the semi­major axis of the WGS84 ellipsoid is 𝑎WGS84 = 6378137 m,and of the Bessel­1841 ellipsoid 𝑎Bessel = 6377397.155 m (see Table 6.1). Hence, the differ­ence is 739.845 m. That is a nearly a kilometer difference at the equator!

1OGP Publication 373­7­2, Geomatics Guidance Note number 7, part 2, June 2013, http://www.epsg.org

7Gravity and gravity potential

In this chapter we introduce, as a preparation for the next chapter, on vertical referencesystems, the concepts of gravity and gravity potential. These concepts are illustrated by meansof the very simple example of the Earth being a perfect sphere. Eventually we introduce thegeoid.

7.1. IntroductionGravity, the main force experienced on Earth, causes (free) objects to change their positions, asto decrease their potential. According to the first two laws by Newton, the force prescribes theacceleration of the object, and this acceleration is the second time derivative of the position.And when there is no resulting force acting on an object, it is not subject to any accelerationand the object will either remain in rest, or be in uniform motion (constant velocity) along astraight line.

The two basic elements of Newtonian mechanics are mass and force. As introduced withTable A.1, mass is an intrinsic property of an object that measures its resistance to acceleration.

In an inertial coordinate system, Newton’s second law states that the (net external) force(vector) 𝐅 equals mass 𝑚 times acceleration (vector)

𝐅 = 𝑚 (7.1)

The acceleration vector is the second derivative with respect to time of the position co­ordinates vector 𝐱. The unit of the derived quantity force is is Newton [N] and equals [kgm/s2].

7.2. Earth’s gravity fieldThe Earth’s gravitational field, represented by acceleration vector 𝐠 (with direction and mag­nitude), varies with location on Earth, as well as above the Earth’s surface. This accelerationvector has special symbol 𝐠, instead of the general 𝐚 or . Acceleration has unit [m/s2].

Weight is defined as the force of gravity on an object. Its magnitude is (also) not aconstant value over the Earth or above it. If the force of gravity is the only one force acting onan object, the object is said to be in free fall with acceleration 𝑔 = ‖𝐠‖ and its apparent weightis zero. A satellite orbiting the Earth is (in the ideal situation) in free fall; the acceleration 𝐠,directed towards the center of Earth, makes the satellite maintaining the circular orbit. Anobject located extremely far from the Earth (and any other body) would be truely weightless(but still have the same mass).

51

52 7. Gravity and gravity potential

Figure 7.1: The (size of) acceleration −𝑔 (left) and the potential𝑊 (right) both approach zero as radial distance 𝑟goes to infinity. A radial distance of zero corresponds to the Earth’s center. The curves start at the Earth’s surface(the thin vertical line at 𝑟 = 6378 km). The dashed line indicates the radial distance of the orbit of a GPSsatellite; the acceleration (across track) is less than 1 m/s2, at a speed of almost 4 km/s. The negative sign forthe acceleration, −𝑔, is used to match the radial direction for an object in free fall.

The magnitude of weight is given by a spring scale. The spring is designed to balance theforce of gravity. The spring scale converts force (in [N]) into mass (in [kg]) on the display byassuming a magnitude of 𝑔 equal to 9.81 m/s2, (approximately everywhere) on and near theEarth’s surface.

For an ideal spherical(ly layered) Earth (or when all of its mass were concentrated at itscenter), the gravitational force exerted on an object with mass 𝑚 a distance 𝑟 away, is givenby

𝐅 = −𝐺𝑀𝑚𝑟2𝐫𝑟 (7.2)

where 𝐺 is the universal gravitational constant (𝐺 = 6.6726 ⋅ 10−11 Nm2/kg2), 𝑀 the mass ofthe Earth (𝑀 ≈ 5.98 ⋅1024 kg), 𝐫/𝑟 the unit direction vector from the Earth’s center toward theobject, and 𝑟 the (radial) distance from the Earth’s center to the object outside the Earth. Theforce (vector) 𝐅, exerted by the Earth on the object with mass 𝑚 is directed from the objecttoward the Earth’s center.

The gravitational attraction consequently reads

𝐠 = −𝐺𝑀𝑟2𝐫𝑟 (7.3)

The force in Eq. (7.2) has been divided by mass𝑚 to obtain the acceleration in Eq. (7.3), whichconsequently could be interpreted as the force per unit mass. The magnitude of the gravi­tational acceleration 𝑔 = ‖𝐠‖ in Eq. (7.3) decreases with increasing height above the Earth’ssurface, and reduces to zero at infinite distance, see Figure 7.1 (at left). The acceleration(vector) 𝐠 also points toward the Earth’s center.

7.3. Gravity PotentialThe work done by a force equals the in­ or dot­product of the force vector 𝐅 and the displace­ment vector 𝑑𝐬, according to

𝑊 = ∫𝐵

𝐴𝐅 ⋅ 𝑑𝐬 (7.4)

7.4. Geoid 53

The (tangential component of the) force is integrated along the path travelled by the objectfrom 𝐴 to 𝐵. Work, a scalar, is expressed in joule [J], the unit of energy, and equals J = Nm.

Taking the force per unit mass in Eq. (7.4), which is actually the acceleration in Eq. (7.3),and interpreting the work in Eq. (7.4) as a difference in energy Δ𝑊 = 𝑊𝐵−𝑊𝐴 after and beforethe force carrying the object from 𝐴 to 𝐵, causes 𝑊 in Eq. (7.4) to be the potential energyper unit mass of the gravitational force, or the potential of gravitation for short. The word‘potential’ expresses that energy can be, but not necessarily is, delivered by the force.

With 𝑚 = 1 kg in Eqs. (7.2)) and (7.4), the potential of gravitation becomes

𝑊 = 𝐺𝑀𝑟 (7.5)

for a pointmass or spherical Earth. In this (geodetic) sense, the potential is expressed in[Nm/kg], which equals [m2/s2]. The potential is a function of the radial coordinate 𝑟, andthe integration constant has been chosen such that the potential is zero at infinite distance,see also Figure 7.1 at right. Substituting Eq. (7.5) in Eq. (7.3), gives the relation betweengravitational acceleration and gravitational potential

𝐠 = −𝑊𝑟𝐫𝑟 (7.6)

with for the magnitude 𝑔 = ‖𝐠‖ = 𝑊𝑟 . Also the derivative of the potential of gravitation

(with respect to position) is roughly equal to the gravitational acceleration, i.e. 𝑑𝑊𝑑𝑟 ≃ 𝑔 (for ahomogeneous sphere the relation is exact), in other words, the slope of the curve at right inFigure 7.1 equals the acceleration shown at left.

From a physics perspective𝑊 in Eq. (7.5) presents the work done per unit mass. In physicsit is common practice to define the potential energy function (with symbol 𝑈), such that thework done by a conservative force equals the decrease in the potential energy function (thatis, use an opposite sign, for instance in Eq. (7.5)).

7.4. GeoidAccording to Eq. (7.5), the potential 𝑊 is constant 𝑊 = 𝑊𝑜, when radial distance 𝑟 is con­stant, that is, on spherical surfaces around the Earth, all centered at the middle of the Earth.Surfaces where the gravity potential 𝑊 is constant are equipotential surfaces, and the grav­ity vector 𝐠 is everywhere orthogonal to them (dictating the local level, according to whichgeodetic instruments are set up). The surface of reference in a vertical sense for physicalphenomena on Earth like water flow is the geoid, the equipotential surface at mean sea level(MSL).

A geoid is shown in Figure 7.2. Obviously, good knowledge of the geoid is crucial forcoastal engineering and construction of canals. On a global scale, the Earth GravitationalModels (EGMs) are the most commonly used geopotential models of the Earth. They consistof spherical harmonic coefficients published by the US National Geospatial­Intelligence Agency(NGA) with reference to the GRS80–ellipsoid that is also used by WGS84 and ITRS (https://earth­info.nga.mil/GandG/update/index.php?dir=wgs84&action=wgs84#tab_egm2008). Three versions of EGM are published: EGM84 with degree and order of harmoniccoefficients 180, EGM96 with degree and order 360, and EGM2008 with degree and order2160. The higher the degree and order of harmonic coefficient, the more parameters themodels have, and the more precise they are. Also provided by NGA is a 2.5­minute world­wide geoid height file, precomputed from the EGM2008. The first EGM, EGM84, was definedas a part of WGS84, and is still used by many GPS devices to convert ellipsoidal height into

54 7. Gravity and gravity potential

Figure 7.2: The height of the geoid with respect to the best fitting Earth ellipsoid (GRS80). The color­scale ranges from about ­100 m (blue) to +70 m (red). This geoid is based on GRACE data. Imagefrom ESA (http://www.esa.int/spaceinimages/Images/2004/10/The_Earth_s_gravity_field_geoid_as_it_will_be_seen_by_GOCE).

height above mean sea­level. The resolution and precision of global models is not sufficientfor applications on a local scale. Therefore, many countries, including the Netherlands (seeSection 10.3), have computed more precise geoids over a smaller region of interest.

As an introduction, the (shape of the) Earth and its gravity field have been treated here asbeing (perfectly) spherical, just like in Section 4.1. Reality (and an adequate model thereof)is much more complicated. As a second approximation the Earth is taken to be a rotationalellipsoid (oblateness of the Earth) as in Section 4.2, and subsequently the inhomogeneousdistribution of mass within (and on) the Earth, and the presence of heavenly bodies are con­sidered. Hence, the shape of the geoid, in particular departures from being a sphere or anellipsoid, is determined by the actual mass distribution of the Earth, the outside surface shape,and also inside. The shape of the geoid may vary over time, think for instance of mass loss inpolar regions due to ice and snow melt, sea­level rise and ground water level changes.

The acceleration experienced on Earth in practice (and hence observable) consists of, first,gravitational acceleration due to the mass of Earth, as discussed above, but also of Sun andMoon (tidal acceleration) and secondly, centrifugal acceleration due to the Earth’s rotations(this effect is largest at the equator, and absent at the poles). Two additional contributions arethe inertial acceleration of rotation and the Coriolis acceleration, which is absent if the object(or measurement equipment) is in rest, or in free fall. Gravity is then commonly defined asthe sum of gravitational acceleration, but discounting the part due to the attraction of Sunand Moon, and centrifugal acceleration.

7.5. GravimetryWith levelling, increments (distances) are measured along the (local) direction of gravity; the(vertical) center line of the instrument is a tangent line to the geoid. Gravity determinesthe direction of the height system; the plane perpendicular to the vector of gravity (locally)represents points at equal height.

The purpose of gravimetry is to eventually describe the geoid with respect to a chosen(geometric) reference body, for instance a rotating equipotential ellipsoid. It comes to deter­mination of the geoid height.

The Earth gravity potential𝑊 itself (in an absolute sense) can not be observed. Inferences

7.6. Conclusion 55

sphere ellipsoid geoid Earth’s surface< 25 km reference < 150 m < 10 km

Table 7.1: Deviations of different (best fitting) models of the Earth, and also the actual Earth’s surface (topography),all referenced to the shape of the ellipsoid.

about the potential have to be made through measurements mainly of the first order (posi­tional) derivative, that is through the gravity vector 𝐠 of which direction and magnitude canbe observed. The direction of gravity can be observed by astronomical measurements (lati­tude and longitude). The magnitude of gravity can be observed by absolute measurements(a pendulum or a free falling object), or by relative measurements (with a spring gravimeter).

At the Earth’s surface the magnitude of gravity changes by 3 ⋅ 10−6 m/s2 over a 1 meterheight difference (𝑑𝑔𝑑𝑟 ). This is the slope of the curve in the graph at left in Figure 7.1,

A satellite falling around the Earth can also be looked upon as an accelerometer, as itsorbit is primarily governed by the Earth’s gravity field.

Gradiometers measure second order (positional) derivatives of the gravity potential, forinstance in a satellite by two (or more) accelerometers at short distance. They sense thedifference in acceleration (differential accelerometry). A satellite tandem mission, where twosatellites closely go together, has a similar purpose.

Finally it should be noted that strictly speaking the separation made between geometricand (physical) gravimetric observables is not a distinct one. They are unified in the theory ofgeneral relativity: the path of a light ray for instance (as used for electronic measurements ofdistance) will bend as it travels through a (strong) gravity field.

7.6. ConclusionIn this chapter we learned that surfaces where the gravity potential𝑊 is constant are equipo­tential surfaces. The gravity vector 𝐠 is everywhere orthogonal to them, dictating the locallevel, and hence water flow. The equipotential surface at mean sea level (MSL), the geoid, isthererfore the ideal surface of reference in a vertical sense.

A rotational ellipsoid (oblateness of the Earth), as in Section 4.2, is a reasonable approxi­mation to the Earth’s geoid. This approximation is popular when not specifically dealing withphysical heights and the flow of water. The deviations between the geoid and rotational ellip­soid are smaller than 150 meters, as shown in Figure 7.2 and Table 7.1. Table 7.1 also includesthe deviation with topography and a spherical approximation of the Earth.

The shape of the geoid, in particular departures from being a sphere or an ellipsoid, isdetermined by the actual mass distribution of the Earth. The shape of the geoid may varyover time, think for instance of mass loss in polar regions due to ice and snow melt, sea­levelrise and ground water level changes.

7.7. Problems and exercisesQuestion 1 Compute the magnitude of the acceleration due to attraction by the Earth’s mass,at the equator, and at a pole, assuming the Earth is a perfect ellipsoid (WGS84), and all massis concentrated in the Earth’s center.Answer 1 The acceleration due to attraction by the Earth is given by Eq. (7.3), whichholds for a spherical Earth with its mass homogeneously distributed, or all mass concen­trated in the Earth’s center. ‖𝑔‖ = 𝐺𝑀

𝑟2 , this is the magnitude of the gravitational acceler­ation at radius 𝑟 away from the Earth’s center. The Earth’s gravitational constant is 𝐺𝑀 =3986004.418 ⋅ 108m3/s2 (Table 6.1). At the equator the distance to the Earth’s center equals

56 7. Gravity and gravity potential

𝑎 = 6378137.0 m (Table 6.1, semi­major axis of WGS84 ellipsoid), and at a pole 𝑏 =6356752.314 m, see question 3 in Section 4.7. Hence the acceleration at the equator (with𝑟 = 𝑎) is ‖𝑔‖ = 9.798m/s2, and at a pole (with 𝑟 = 𝑏) is ‖𝑔‖ = 9.864m/s2.

Question 2 As a follow­up on question 1, compute the (magnitude of the) centrifugal accel­eration at the equator.Answer 2 The magnitude of the centrifugal acceleration follows from the velocity and theradius: 𝑎 = 𝑣2

𝑟 (uniform circular motion). Hence, we need the velocity. The Earth makes a fullturn in (one solar day) of 𝑇 = 23h56m = 86160 seconds. At the equator the circumferenceis 2𝜋𝑎 (with the radius set equal to the length of the semi­major axis 𝑎, not to be confusedwith the symbol for acceleration which is used later), and hence velocity 𝑣 is 𝑣 = 2𝜋𝑎/𝑇 =465.1m/s. The acceleration becomes 𝑎 = 0.034m/s2. The centrifugal acceleration is pointingoutward. The acceleration due to the attraction by the Earth’s mass is pointing inward to thecenter of the Earth. At a pole, the centrifugal acceleration is zero.

Question 3 Suppose again that the Earth is a perfect ellipsoid, and that all mass is concen­trated in the Earth’s center. Would water flow from the equator to the poles, in case the Earthwould be not rotating?Answer 3 Water flow is dictated by potential. The Earth is not rotating, hence we needto consider only the gravitational potential, due to the attraction by the Earth’s mass. Theequation for potential is simply𝑊 = 𝐺𝑀

𝑟 (7.5), at a location 𝑟 away from the Earth’s center. Atthe equator we have 𝑟 = 𝑎 (the length of the semi­major axis of the ellipsoid), and at a pole𝑟 = 𝑏 (the length of the semi­minor axis). As 𝑏 < 𝑎, we have 𝑊pole > 𝑊equator. The potentialis zero at 𝑟 = ∞, and the potential is larger at the pole (than at the equator), hence in thiscase water would flow from the equator to the poles.

8Vertical reference systems

Until now the focus has been on the geometry of points on the Earth surface (location), usingfor instance geographic latitude and longitude on a reference ellipsoid, or x­ and y­coordinatesin a map projection. Now it is time to turn our attention to specifying the height, or elevation,of points.

8.1. Ellipsoidal heightsThe elevation of a point can only be expressed with respect to another point or referencesurface. In theory, it is possible to use the radius to the Center of Mass (CoM) of the Earth ­also the origin of most 3D coordinates systems ­ as a measure for elevation. This is howeveronly practical for Earth satellites, but not very practical for points on the surface of the Earth.Instead it will be much more convenient to use the height above a reference ellipsoid, as wehave seen in Section 4.2, or to use a different ­ more physical ­ definition of height.

Ellipsoid

Geoid (MSL)

Earth surfaceH

N

h = N + H

h : Ellipsoidal height (h=N+H)H : Orthometric heightN : Geoid height

Sea surface h

Figure 8.1: Relation between ellipsoidal height ℎ, orthometric height 𝐻 and geoid height 𝑁.

The main drawback of ellipsoidal height is that surfaces of constant ellipsoidal height arenot necessarily equipotential surfaces. Hence, in an ellipsoidal height system, it is possiblethat water flows from a point with low ’height’ to a point with a higher ’height’. This defiesone on the main purposes of height measurements: defining water levels and water flow.Also, ellipsoidal heights are a relatively new concept, which can only be measured using spacegeodetic techniques such as GPS. Since heights play an important role in water management

57

58 8. Vertical reference systems

and hydraulic engineering a necessary requirement is that water always flows from a pointwith a higher height to points with lower height.

8.2. Orthometric and normal heightsIn fact, instead of working with heights and height differences which are expressed in meters,it is actually more appropriate to use the gravitational potential𝑊 or potential differences Δ𝑊.For a discussion of gravity and potential numbers the reader is referred to Chapter 7. Here itsuffices to recapitulate from Chapter 7 that an equipotential surface, with potential 𝑊0 suchthat it more or less coincides with mean sea­level over Sea, fulfills all the requirements for areference surface for the height. The unit of potential 𝑊 is [Nm/kg] which equals [m2/s2].From Eq. (7.3) and (7.5) follows that the potential difference Δ𝑊 and height difference Δ𝐻are related,

Δ𝑊 = −𝑔 Δ𝐻 (8.1)

with 𝑔 the gravitational acceleration in [m/s2]. The gravitational acceleration is a positivenumber: the minus sign in Eq. (8.1) is because the gravitational potential 𝑊 decreases withincreasing height, see the right of Figure 7.1, and therefore Δ𝑊 and Δ𝐻 have opposite sign.Note that the gravitational acceleration 𝑔 is not constant but depends on the location on Earthand height. Orthometric heights are defined by the inverse of Eq. (8.1),

𝐻orthometric = −1𝑔(𝑊 −𝑊0) . (8.2)

with𝑊0 the potential of the chosen reference equipotential surface. Normal heights are basedon the normal gravity 𝛾 instead of the gravitational acceleration 𝑔,

𝐻normal = −1𝛾(𝑊 −𝑊0) (8.3)

with 𝛾 the normal gravity from a normal (model) gravity field that matches gravitational accel­eration for a selected reference ellipsoid with uniform mass equal to the mass of the Earth. Inorder to distinguish orthometric and normal heights from ellipsoidal heights we use a capital𝐻 for orthometric and normal heights, and a lower case ℎ for ellipsoidal heights.

The relation between orthometric (normal) height 𝐻 and ellipsoidal height ℎ is given bythe following approximation

ℎ = 𝑁 + 𝐻 (8.4)

with 𝑁 the height of the geoid above the ellipsoid. This is illustrated in Figure 8.1. Thisapproximation is valid near the surface of the Earth. In fact, some of the smaller effects, orthe difference between normal and orthometric height, are often lumped with the geoid heightinto 𝑁, which then strictly speaking is a correction surface for transforming orthometric (ornormal) height to ellipsoidal heights.

Instead of the word ‘height’, the word ’elevation’ is often used in the US, to refer to theheight of a point on the Earth’s surface above a geoid.

8.3. Height datumsThe zero point, or datum point, for the heights depends on the choice of𝑊0. This datum pointis often defined based on tide­gauge data such that the geoid is close to mean sea­level (MSL).In Figure 8.2 the reference tide­gauges used for different European countries are shown,

8.3. Height datums 59

Figure 8.2: Differences between national height datums for Europe and reference tide­gauges in centimeters(Source: BKG http://www.bkg.bund.de).

together with the difference in the height datums. The differences have been computed fromthe European re­adjustment of precise levellings. The effects of using different tide­gauges,and the differences between mean sea­level for the North Sea, Baltic Sea, Mediteranian andBlack Sea are clearly visible. Also some countries, for instance Belgium, did not use meansea­level to define their height datum but used low water spring as reference.

It is not necessary to use mean sea­level as reference surface for all applications. Inparticular for hydrography, it is more common to use the Lowest Astronomical Tide (LAT) asreference surface. This is not an equipotential surface as this reference surface also dependson the tidal variations.

Lowest Astronomical Tide (LAT) is the lowest predicted tide level that can occur underany combination of astronomical conditions assuming average meteorological conditions. Theadvantage for Hydrographic chart datums is that all predicted tidal heights must then bepositive, although in practice lower tides may occur due to e.g. meteorological effects. Inthe Netherlands, UK and many other countries charted depths and drying heights on nauticalcharts are given relative to LAT, and tide tables give the height of the tide above LAT. The

60 8. Vertical reference systems

depth of water, at a given point and at a given time, is then calculated by adding the charteddepth to the height of the tide, or by subtracting the drying height from the height of the tide,with all heights and depths given with respect to LAT.

8.4. Problems and exercisesQuestion 1 Modeling the Earth as a sphere with radius equal to the semi­major axis of theWGS84­ellipsoid (see Table 6.1), and assuming that all mass is concentrated in the Earth’scenter, compute the gravitational acceleration.Answer 1 The semi­major axis of the WGS84­ellipsoid is 𝑎 = 6378137 m. The Earth’s gravi­tational constant is 𝐺𝑀 = 3986004.418 ⋅108 m3/s2. The magnitude of the gravitional acceler­ation at radius 𝑟 away from the Earth’s center is simply 𝑎 = 𝐺𝑀/𝑟2, see Chapter 7, Eq. (7.2).This yields 𝑎 = 9.798 m/s2. The acceleration vector points downwards to the Earth’s center.

Question 2 The ellipsoidal height of a geodetic marker on Terschelling is 56.098 m. The ellip­soidal height of a geodetic marker in Eijsden, near Maastricht is 103.797m. These coordinatesare given in ETRS89 (and hence based on the WGS84­ellipsoid). The geoid­height­differencebetween these two locations is 4.601m (that is, the geoid height near Maastricht is larger thanin Terschelling; NLGEO2004 geoid with respect to the GRS80/WGS84 ellipsoid). Compute theorthometric (level) height difference between Terschelling and Eijsden.Answer 2 The relation between ellipsoidal height ℎ and orthometric (levelled) height 𝐻 isℎ = 𝐻 + 𝑁, with 𝑁 the geoid height. This relation can also be exploited in a heigh­differenceℎ𝑇𝐸 = 𝐻𝑇𝐸 + 𝑁𝑇𝐸, with ℎ𝑇𝐸 = ℎ𝐸 − ℎ𝑇, with 𝑇 for Terschelling, and 𝐸 for Eijsden. Theellipsoidal height difference between Terschelling and Eijsden is ℎ𝑇𝐸 = ℎ𝐸 − ℎ𝑇 = 47.699 m.The geoid­height difference was given as 𝑁𝑇𝐸 = 4.601 m. Hence, 𝐻𝑇𝐸 = 43.098 m. Hence,the levelled height difference is about 4.5 m smaller than the ellipsoidal height difference.With the levelled height of Terschelling being 𝐻𝑇 = 14.695 m, the levelled height of Eijsdenbecomes 𝐻𝐸 = 57.793 m.

9International reference systems and

frames

In this chapter a number of common international reference systems and frames is discussed.We start with the well know world wide WGS84 system used by GPS, but quickly shift forcusto the more important International Terrestrial Reference System (ITRS), which is realizedthrough the International Terrestrial Reference Frames (ITRF). Then the focus is shifted to re­gional reference systems and frames, with the European Terrestrial Reference System ETRS89as our prime example.

9.1. World Geodetic System 1984 (WGS84)The USA Department of Defense (DoD) World Geodetic System 1984 (WGS84) is probably byfar the best known global terrestrial reference system. Which is understandable consideringthe popularity of Global Positioning System (GPS) receivers, but it is also somewhat surprisingconsidering the fact that WGS84 is primarily a US military system.

For civilian users WGS84 coordinates are only obtainable through the use of GPS. The onlyWGS84 realization available to civilian users are the GPS broadcast satellite orbits as civilianusers have no direct access to tracking sites or tracking data from the US military. This meansthat for civilian users the accuracy of WGS84 is restricted to the accuracy of the GPS broadcastorbits, which is of the order of a few meters. Users may try to improve the accuracy to a fewdecimeters by taking averages of GPS station positions over several days, but then if accuracyis really an issue it would be much better to switch to ITRF or ETRS89, discussed in the nextsections.

In fact there are different WGS84 realizations. Until GPS week G7301, WGS84 was based onthe US Navy Doppler Transit Satellite System. Newer realizations of WGS84 are coincident withInternational Terrestrial Refernce Frame (ITRF), see Section 9.2, at about the 10­centimeterlevel. For these realizations there are no official transformation parameters. The newer real­izations are adjusted occasionally in order to update the tracking station coordinates for platevelocity. These updates are identified by the GPS week, i.e. WGS84(G730, G873 and G1150).

In general WGS84 is identical to the ITRS and its realizations ITRFyy at the one meterlevel. Therefore, in practice, when precision does not really matter and the user is satisfiedwith coordinates at the one meter level, coordinates in ITRS, or derivatives of ITRS (like theEuropean ETRS89) are sometimes simply referred to as ”WGS84”.

1The GPS week number is the number of weeks counted since January 6, 1980.

61

62 9. International reference systems and frames

The use of WGS84 should be avoided for applications other than for hiking, regular naviga­tion, and other non­precision applications. The WGS84 is not suited for applications requiringdecimeter, centimeter or millimeter accuracy. For surveying and geoscience applications themore accurate ITRF, or ETRS89 in Europe, should be used.

9.2. International Terrestrial Reference System and FramesThe International Terrestrial Reference System (ITRS) is a global reference system co­rotatingwith the Earth. It is realized through International Terrestrial Reference frames, which providescoordinates of a set of points located on the Earth’s surface.

It can be used to describe plate tectonics, regional subsidence or displacements in a globalcontext, or to represent the Earth whenmeasuring its rotation in space. The ITRF is maintainedthrough an international network of space geodetic observatories and an international networkof GNSS (GPS) tracking stations. The ITRF is the most accurate terrestrial reference frameto date. Therefore, it is frequently used as the basis for other reference frames, or, as anintermediate to describe relations between coordinate systems. For instance, the well knownWGS84, used by GPS, is directly linked to the ITRF.

The definition of the International Terrestrial Reference System (ITRS) is based on IUGG(International Union of Geodesy and Geophysics) resolution No 2 adopted in Vienna, 1991.As a consequence, the ITRS is

• a geocentric co­rotating system, with the center of mass being defined for the wholeEarth, including oceans and atmosphere,

• the unit of length is the meter and its scale is consistent with the Geocentric CoordinateTime (TCG) by appropriate relativistic modeling,

• the time evolution of the orientation is ensured by using a no­net­rotation condition withregards to horizontal tectonic motions over the whole Earth,

• the initial orientation is given by the Bureau International de l’Heure (BIH) orientationat 1984.0.

The ITRS is realized through a number of International Terrestrial Reference Frames (ITRF).Realizing a global terestrial reference system is not trivial as the Earth is not a rigid body. Eventhe outer layer, the Earth’s crust, is flexible and changes under the influence of solid Earthtides, loading by the oceans and atmosphere, and tectonics. From a global perpective pointsare not stationary, but moving. Therefore each individual ITRF contains station positions andvelocities, often together with full variance matrices, computed using observations from spacegeodesy techniques2. The stations are located on sites covering every continent and tectonicplate on Earth. To date there are thirteen realizations of the ITRS 3: ITRF2008 and ITRF2014are the latest two realizations. ITRF2000 will be the next realization, using more recent data,reprocessing of old data, improved models and processing softwares.

The realization of the ITRS is an on–going activity resulting in periodic updates of the ITRFreference frames. These updates reflect

• improved precision of the station positions 𝐫(𝑡0) and velocities due to the availabilityof a longer time span of observations, which is in particular important for the velocities,

2Very Long Baseline Interferometry (VLBI), Lunar Laser Ranging (LLR), Satellite Laser Ranging (SLR), GlobalPositioning System (GPS) and Doppler Orbitography and Radiopositioning Integrated by Satellite system (DORIS).3ITRF88, ITRF89, ITRF90, ITRF91, ITRF92, ITRF93, ITRF94, ITRF96, ITRF97, ITRF2000, ITRF2005, ITRF2008and ITRF2014. The numbers in the ITRF designation specify the last year of data that were used. For examplefor ITRF97, which was published in 1999, space geodetic observations available up to and including 1997 wereused, while for ITRF2000 an additional three years of observations, up to and including 2000, were used.

9.2. International Terrestrial Reference System and Frames 63

Figure 9.1: ITRF2008 velocity field with major plate boundaries shown in green (Figure from http://itrf.ensg.ign.fr/).

• improved datum definition due to the availability of more observations and better models,

• discontinuities in the time series due to earthquakes and other geophysical events,

• newly added and discontinued stations,

• and occasionally a new reference epoch 𝑡0.All ITRF model the (secular) changes in the Earth’s crust. The position 𝐫(𝑡) at a specific epoch𝑡 is given by

𝐫(𝑡) = 𝐫(𝑡0) + ⋅ (𝑡 − 𝑡0) (9.1)

The ITRF2008 velocities are given in Figure 9.1. The velocities, in the vector , are of theorder of a few centimeters per year up to a decimeter per year for some regions. This meansthat for most applications velocities cannot be ignored. It also implies that when coordi­nates are distributed it is equally important to provide the epoch of observation to which thecoordinates refer. Higher frequencies of the station displacements, e.g. due to solid Earthtides and tidal loading effects with sub–daily periods, can be computed using models specifiedin the IERS conventions, chapter 7 (http://www.iers.org/IERS/EN/DataProducts/Conventions/conventions.html).

In Figure 9.2 a time series of station positions for a GPS receiver in Delft is shown. Thetop figure shows a couple of features: (1) the secular motion of the point, (2) jumps in thecoordinate time series and velocity whenever a new ITRF is introduced, and (3) discontinuitiesdue to equipment changes (mainly antenna changes). The bottom figure shows the timeseries after reprocessing in the most recent reference frame. The jumps due to changes inthe reference frame have disappeared and the day­to­day repeatability has been improvedconsiderably due to improvements in the reference frame and processing strategies. Onefeature is not shown in Figure 9.2, and that is the effect of earthquakes. Stations whichare located near plate­boundaries would experience jumps and post­seismic relaxation effectsin the time series due to earthquakes. Although geophysically very interesting, this makesstations near plate boundaries less suitable for reference frame maintenance. Figure 9.3 showsthe same time series, but in the European ETRF2000 reference frame, which is discussed inSection 9.3.

64 9. International reference systems and frames

Figure 9.2: Time series of station positions of a permanent GPS receiver in Delft from 1996­2014. The top figureshows the time series in the ITRFyy reference frame that was current at the time the data was collected. Thebottom figure shows the data after re­processing in the IGS05/IGS08 reference frame that is based on the mostrecent ITRF2008 frame. The vertical red lines indicate equipment changes (Figures from http://www.epncb.oma.be/).

The datum of each ITRF is defined in such a way as to maintain the highest degree ofcontinuity with past realizations and observational techniques 4. The ITRF origin and rates are

4The International Terrestrial Reference Frame (ITRF) is maintained by the International Earth Rotation and Ref­erence Systems Service (IERS). For more information see also http://itrf.ensg.ign.fr/. The IERS is alsoresponsible for the International Celestial Reference Frame (ICRF) and the Earth Orientation Parameters (EOPs)that connect the ITRF with the ICRF. The observational techniques are organized in services, such as the Interna­tional GNSS Service (IGS), International Laser Ranging Service (ILRS) and International VLBI Service (IVS). Forinstance, the IGS is a voluntary organization of scientific institutes that operates together a tracking network ofover 300 stations, several analysis and data centers, and a central bureau. The main product of IGS are preciseorbits for GNSS satellites (including GPS), satellite clock errors and station positions all in the ITRF.

9.3. European Terrestrial Reference System 1989 (ETRS89) 65

essentially based on the Satellite Laser Ranging (SLR) time series of Earth orbiting satellites.The ITRF orientation is defined in such a way that there are null rotation parameters and nullrotation rates with respect to ITRF2000, whereas for ITRF2000 is no net rotation with respectto the NNR­NUVEL1A plate tectonic model is used. These conditions are applied over a corenetwork of selected stations. The ITRF scale and scale rate are based on the VLBI and SLRscales/rates. The role of GPS in the ITRF is mainly to tie the sparse networks of VLBI and SLRstations together, and provide stations with a global coverage of the Earth.

Although the goal is to ensure continuity between the ITRF realizations as much as possiblethere are transformations involved between different ITRF that reflect the differences in datumrealization. Each transformation consists of 14 parameters, a 7­parameter similarity transfor­mation for the positions involving a scale factor, three rotation and three translations, and a7­parameter transformation for the velocities involving a scale rate, three rotation rates andthree translation rates. The translation parameters and formula are published on ITRS websitehttp://itrf.ensg.ign.fr/doc_ITRF/Transfo­ITRF2014_ITRFs.txt. The trans­formation formula is essentially Eq. (3.13), except for a different sign of the transformationparameters.

Web­based Precise Point Positioning (PPP) services, which utilize satellite orbits and clocksfrom the International GNSS Service (IGS), allow GPS users to directly compute positions inITRF. At the same time many regional and national institutions have densified the IGS networkto provide dense regional and national networks of station coordinates in the ITRF.

When working with the ITRF it is typical to provide coordinates as Cartesian coordinates.However, the user is free to convert these into geographic coordinates. The recommendedellipsoid for ITRS is the GRS80 ellipsoid, see Table 6.1. This is the same ellipsoid as used forinstance by WGS84.

9.3. European Terrestrial Reference System 1989 (ETRS89)The European Terrestrial Reference System 1989 (ETRS89) is the standard coordinate systemfor Europe. It is the reference system of choice for all international geographic and geodynamicprojects in Europe5. The system also forms the backbone for many national reference systems.Although the ITRS plays an important role in studies of the Earth’s geodynamics it is lesssuitable for use as a European georeferencing system. This is because in ITRS all points inEurope exhibit a more­or­less similar velocity of a few centimeters per year, as was shown inFigures 9.1 and 9.2.

The ETRS89 terrestrial reference system is coincident with ITRS at the epoch 1989.0 andfixed to the stable part of the Eurasian Plate. The year in the name ETRS89 refers explicitlyto the time the system was coincident with ITRS6. ETRS89 is accessed through the EUREFPermanent GNSS Network (EPN) 7, a science­driven network of continuously operating GPSreference stations with precisely known station positions and velocities in the ETRS89, orthrough one of many national or commercial GPS networks which realize ETRS89 on a nationalscale.

5The ETRS89 was established in 1989 and is maintained by the sub­commission EUREF (European ReferenceFrame) of the International Association of Geodesy (IAG). ETRS89 is supported by EuroGeographics and endorsedby the European Union (EU).6Sometimes people think the coordinates should be given at epoch 1989.0, but this is not necessary, as coordinatescan be given at any epoch. The best practice is to give the coordinates at the epoch of observation.7All contributions to the EPN are voluntary, with more than 100 European agencies and universities involved. Thereliability of the EPN network is based on extensive guidelines guaranteeing the quality of the raw GPS data to theresulting station positions and on the redundancy of its components. The GPS data is also used for a wide rangeof scientific applications such as the monitoring of ground deformations, sea level, space weather and numericalweather prediction.

66 9. International reference systems and frames

Figure 9.3: Time series of station positions of a GPS receiver in Delft from 1996­2014 in the European referenceframe ETRF2000. The horizontal station velocity In ETRS89 is at the few mm level. The vertical red lines indicateequipment changes which cause jumps of a few mm in the time series (Figure from http://www.epncb.oma.be/).

Station velocities in ETRS89 are generally very small because ETRS89 is fixed to the stablepart of the Eurasian plate. Compared to ITRS, with station velocities in the order of a fewcentimeter/year, station velocities in ETRS89 are typically smaller than a few mm/year. This isclearly illustrated in Figure 9.3 for the GPS station in Delft that was also used for Figure 9.2.Of course, there are exceptions in geophysically active areas, but for most practical applica­tions, one may ignore the velocities. This makes ETRS89 well suited for land surveying, highprecision mapping and Geographic Information System (GIS) applications. Also, ETRS89 iswell suited for the exchange of geographic data sets between European national and interna­tional institutions and companies. On other continents solutions similar to ETRS89 have beenadopted.

The ETRS89 system is realized in several ways, and like with ITRS, realizations of a sys­tem are called reference frames. By virtue of the ETRS89 definition, which ties ETRS89 toITRS at epoch 1989.0 and the Eurasian plate, for each realization of the ITRS (called ITRFyy),also a corresponding frame in ETRS89 can be computed. These frames are labelled ETRFyy.Each realization has a new set of improved positions and velocities. The three most recentrealizations of ETRS89 are ETRF2000, ETRF2005 and ETRF20148. Since each realization alsoreflects improvements in the datum definition of ITRF, which results in small jumps in thecoordinate time series, the EUREF Governing Board (formally Technical Working Group) rec­ommends not to use the ETRF2005 for practical applications, and instead to adopt ETRF2000as a conventional frame of the ETRS89 system. However, considering the diverse needs ofindividual countries’, it is the countries’ decision to adopt their preferred ETRS89 realization.Most countries adopted the recommended ETRF2000, but not every European country has,and considering the improved accuracy and stability of the ITRF2014, some could switch toETRF2014.

Another way to realize ETRS89 is by using GNSS campaign measurements or a network

8There is no ETRF2008.

9.4. Problems and exercises 67

of permanent stations. From 1989 onwards many national mapping agencies have organizedGPS campaigns to compute ETRS89 coordinates for stations in their countries, and then linktheir national networks to ETRS89. Later on these campaigns were replaced by networks ofpermanent GPS receivers. These provide users with downloadable GPS data and coordinatesin ETRS89 that they can use together with their own measurements. The permanent networksalso provide 24/7 monitoring of the reference frames. An example is the Active GPS ReferenceSystem for the Netherlands (AGRS.NL), which was established in 1997. Nowadays the DutchKadaster, as well as several commercial providers, operate real­time network RTK services(NETPOS, 06­GPS, LNRNET, and others) that provide GPS data and corrections in real­timewhich allows instantaneous GPS positioning in ETRS89 at the centimeter level. The Dutchnetwork RTK services are certified by the Dutch Kadaster and thus provide a national realizationof ETRS89 linked to the ETRF2000 reference frame. Similar services are operated in manyother European countries.

When working with ETRS89 it is typical to provide coordinates as either Cartesian coor­dinates, or, geographic coordinates and ellipsoidal height (using the GRS80 ellipsoid). Thereis no European standard for the type of map projection to be used, so the user can stillselect a favorite map projection depending on the application at hand. However, EuroGeo­graphics does recommend to use one of three selected projection: Lambert Azimuthal EqualArea (ETRS89/LAEA, EPSG:3035) for statistical mapping at all scales and other purposeswhere true area representation is required, Lambert Conformal Conic 2SP (ETRS89/LCC,EPSG:3034) for conformal mapping at 1:500,000 scale or smaller, or Universal TransverseMercator (UTM) for conformal mapping at scales larger than 1:500,000. In several coun­tries, including the Netherlands, a conventional transformation from ETRS89 to grid (map)coordinates that resemble the old national systems is provided, see Chapter 10. A serviceto convert coordinates from ITRS to ETRS89, and vice versa, is provided at http://www.epncb.oma.be/_productsservices/coord_trans/. Specifications for the transforma­tion procedure and reference frame fixing can be found here http://etrs89.ensg.ign.fr/pub/EUREF­TN­1.pdf.

9.4. Problems and exercisesQuestion 1 The position coordinates of a geodetic marker in Westerbork are given in theITRF2008 at epoch 2015.0 as 𝑋 = 3828735.710 m, 𝑌 = 443305.117 m, 𝑍 = 5064884.808 m,with velocities 𝑉𝑋 = −0.0153 m/y, 𝑉𝑌 = 0.0160 m/y, 𝑉𝑍 = 0.0096 m/y. Compute the positioncoordinates of this marker, in ITRF2008, for January 1st, 2016.Answer 1 The position coordinates are given in the International Terrestrial Reference Frame2008, a realization of the ITRS. Generally positions are subject to small movements withina global reference system (due to Earth’s dynamics). In this question the coordinates aregiven for January 1st, 2015. And also the velocities are given (in meter per year) to computethe position at any other instant in time. We can propagate the position over 1 year toJanuary 1st, 2016. The resulting coordinates become: 𝑋 = 3828735.695 m, 𝑌 = 443305.133,𝑍 = 5064884.818.

10Dutch national reference systems

In this chapter the Dutch triangulation system RD and height system NAP, and their relationto the ETRS89, are presented. The focus in this chapter is on the the Netherlands, with theremark, that many other countries have undergone similar developments and adopted similarapproaches.

10.1. Dutch Triangulation System (RD)The Dutch Triangulation System (RD), in Dutch Rijksdriehoeksstelsel, has a history dating fromthe 19th Century. Following a century of traditional triangulations, GPS started to replacetriangulation measurements in 1987. The increasing use of GPS resulted in a redefinitionof RD in 2000, whereby from 2000 onwards RD was linked directly to ETRS89 through atransformation procedure called RDNAPTRANS.

Figure 10.1: First order triangulation network for the Netherlands of 1903 (left) and GPS base network of 1997(right). Figure from de Bruijne et al., 2005.

69

70 10. Dutch national reference systems

10.1.1. RD1918The first­order triangulation grid was measured in the years between 1885 and 1904 (Fig­ure 10.1). The church tower of Amerfoort was selected as the origin of the network and asreference ellipsoid the ellipsoid of Bessel (1841) was chosen. The scale was derived from adistance measurement on a base near Bonn, Germany. Between 1896 and 1899 geodetic­astronomic measurements were carried out at thirteen points throughout the Netherlands inorder to derive the geographical longitude and latitude of the origin in Amersfoort and theorientation of the grid. As map projection an oblique stereographic double projection was se­lected (Heuvelink, 1918). The projection consists of a Gauss­Schreiber conformal projectionof Bessel’s ellipsoid (1841) onto a sphere, followed by a oblique stereographic projection ofthe sphere to a tangential plane, as shown in Figure 10.2.

The stereographic projection is a perspective projection from the point antipodal to thecentral point in Amersfoort on a plane parallel to the tangent at Amersfoort. This projectionis conformal, which means the projection is free from angular distortion, and that lines in­tersecting at any specified angle on the ellipsoid project into lines intersecting at the sameangle on the projection. Therefore, meridians and parallels will intersect at 90∘ angles in theprojection, but, except for the central meridian through Amersfoort, meridians will convergeslightly to the North and do not have constant x­coordinate in RD. This is known as meridianconvergence. This projection is not equal­area. Scale is true only at the intersection of theprojection plane with the sphere and is constant along any circle around the center point inAmerfoort. However, by letting the tangential projection plane intersect the sphere, the scaledistortions at the edges of the projection domain will be within reasonable limits.

Figure 10.2: RD double projection (Bessel ellipsoid → Sphere → Plane) and definition of RD coordinates (FigureT.Nijeholt at nl.wikibooks).

During the years between 1898­1928 a densification programme was carried out whichresulted in the publication of 3732 triangulation points. At the time of publication already 365points had disappeared or were disrupted. To prevent further reduction in points and maintainthe network the ”Bijhoudingsdienst der Rijksdriehoeksmeting” was established at the DutchCadastre. From 1960 to 1978 a complete revision was carried out and the RD system wasalso connected to neighboring countries, which resulted in a reference frame with roughly

10.1. Dutch Triangulation System (RD) 71

6000 points at mutual distances of 2.5­4 km. To prevent confusion between the x­coordinatesand y­coordinates, and to obtain always positive coordinates, the origin of the coordinates wasshifted 155 km to the West and 463 km to the South (False Easting and Northing). This resultedin only positive coordinates and y­coordinates that are always larger than the x­coordinates.It also avoids confusion between the old and new coordinates.

Starting in 1993 a so­called GPS base network of 418 points was established to offer GPSusers a convenient way to connect to the RD system. See Figure 10.1. Most of traditionaltriangulation points, many of which are church spires or towers, are not accessible to GPSmeasurements. The points in the GPS base network have an unobstructed view of the skyand are easily accessible by car. The GPS base network points are located at distances of10 to 15 km from each other, which is well suited for GPS baseline measurements. Thepoints in the GPS base network have been connected to neighboring RD points to determineRD coordinates and by second­order levelling to neighboring NAP benchmarks to determineheights. In addition, the point in the GPS base network were connected by GPS measurementsto points in the European ETRS89 system. As a result the GPS base network points havemeasured coordinates both in RD/NAP and ETRS89. This made it possible ­ for the first time­ to study systematic errors in the RD system. It was found that the RD system of 1918 hassystematic errors of up to 25 cm with significant regional correlations, as shown in Figure 10.3for the province of Friesland. In the age of GPS this may seem as a large number, but in 1918this was an excellent accomplishment.

Figure 10.3: Differences between RD and ETRS89 based coordinates for the GPS Kernnet in the province ofFriesland, showing significant regional correlation between the vectors (Figure from de Bruijne et al., 2005).

The systematic errors in the RD system were never an issue until the introduction of GPS.Before GPS, all measurements were connected to nearby triangulation points which could al­most always be found within a radius of 3­4 km, and users never noticed large discrepanciesunless the triangulation points were damaged. However, with GPS it became routine to mea­sure over distances of 15 km up to 100 km to the nearest GPS basenet point or permanentGPS receiver, and then systematic errors in RD will become noticeable. This led to a majorrevision in the definition of RD in 2000.

72 10. Dutch national reference systems

,

NAP H

pseudo RD x’, y’

Bessel (1841) X, Y, Z

RD x, y

Bessel (1841)

RD correction grid

RD map projection

ETRS89

h

ETRS89 X, Y, Z

7-parameter

transformation

Geoïd model

User environment: input & output

Processing environment

h

Figure 10.4: RDNAPTRANS transformation procedure until RDNAPTRANS™2018. The figure outlines the rela­tionships and transformations between ETRS89, RD2000 and NAP (Figure after de Bruijne et al., 2005). Thecoordinates below the line, with the exception of Cartesian coordinates in ETRS89, are used only for computa­tional purposes and should never be published or distributed to other users. In the new RDNAPTRANS™2018procedure a correction grid is applied to the ETRS89 latitude and longitude instead, and pseudo RD coordinatesbecome the final RD coordinates. In the new RDNAPTRANS™2018 there is also a version in which the 7­parametertransformation is included in the correction grid, see Figure 10.5.

10.1.2. RD2000In 2000 a new definition of the RD grid was adopted and assigned the name RD2000. Thisdefinition replaces Heuvelink’s (1918) definition, which is since then referred to as RD1918.

In the new definition RD2000 is based on ETRS89. Within this new definition two typesof coordinates are allowed to be used in practice: (1) Cartesian or geographic coordinates inETRS89, and (2) RD x­ and y­coordinates. The big difference is that the RD coordinates arenow obtained by a conventional transformation from the ETRS89 coordinates. The transfor­mation has been assigned the name RDNAPTRANS. This definition was chosen to minimizethe impact for users. GPS users can happily work with ETRS89, and if they wish, transformtheir coordinates to RD at the very last stage. Owners of large databases with geographicinformation in RD have their investments protected and do not need to make changes.

The new definition has not changed the published RD coordinates significantly. In addition,the European ETRS89 frame was introduced as the three­dimensional reference frame for theNetherlands. This was effected by the publication of the ETRS89 coordinates along with RDcoordinates.

The RDNAPTRANS transformation procedure of Figure 10.4 is an essential part of the

10.1. Dutch Triangulation System (RD) 73

RD2000 definition. It has four main elements:

1. 7­parameter transformation from ETRS89 to an intermediate system defined on theBessel, 1841, ellipsoid, including conversions from Cartesian to geographic coordinates),resulting in latitude, longitude and height on the Bessel, 1841, ellipsoid.

2. a map projection using the same constants and definitions as RD1918, including a falseEasting and Northing of 155 km and 463 km. The projected coordinates are referred toas ”pseudo RD”.

3. a conventional correction grid for the x­ and y­coordinates in RD, which ’corrects’ thepseudo RD coordinates of the previous step for the systematic distortions in the oldRD­grid. The corrections are obtained by interpolation in the correction grid.

4. quasi–geoid for the conversion between NAP heights and height on the GRS80 ellipsoid,which will be discussed next in Section 10.2.

The transformation procedure works in both directions, and, both for 2D and 3D coordinates.In case no heights are available the Kadaster recommends to use an approximate height, e.g.by using a digital terrain model, or, when that is not possible, to use ℎ = 43 m (which is closeto 𝑁𝐴𝑃 = 0) so that one gets the same result after transforming back and forth. In thesecases geographical latitude and longitude can be used, but heights, as well as 3D Cartesiancoordinates are meaningless. Outside the transformation procedure the use of geographiccoordinates on the Bessel­1841 ellipsoid and pseudo­RD coordinates is not recommended.For geographic coordinates solely ETRS89 coordinates should be used within the Netherlands.For RD­coordinates only coordinates that include the systematic distortions should be used.Failing to do so may result in pollution and errors of existing databases based on RD.

Since 2000 two minor revisions of RD2000 occurred in 2004 and 2008, and one majorrevision in 2019. The minor revisions were related to changes in the European referenceframe, which affected the 7­parameter transformation, and the introduction of an improvedNLGEO2004 geoid in 2004. The original 2000 version used the ‘De Min‘ geoid and oldertransformation parameters. The modified transformation procedures are referred to as RD­NAPTRANS™2004 and RDNAPTRANS™2008. The original transformation of 2000 is sincethen also referred to as RDNAPTRANS™2000. In 2018 work started on a major revision ofRDNAPTRANS, resulting in RDNAPTRANS™2018, which was published in 2019. In RDNAP­TRANS™2018 the correction grid is aplied to the latitude and longitude coordinates insteadof the pseudo RD coordinates, the NLGEO2018 quasi–geoid is used instead of NLGEO2004geoid, and the interpolation and correction grids are based on international standards.

More revisions may be possible in the future , as there is a need to maintain a close linkwith the most up to date realizations of ETRS89 as well as to retain as constant as possibleRD coordinates.

10.1.3. RDNAPTRANS™2018Although the RDNAPTRANS transformation procedure is well documented and example sourcecode in C and Matlab is available free of charge, the transformation procedure was only sup­ported by a few Geographic Information System (GIS) packages. The correction grid that wasused in older versions of RDNAPTRANS was often not directly supported by software and themap projection chosen for RD was considered to be exotic. This, combined with the fact thatthe 7­parameter transformation entails a significant shift and rotation, has sparked a discus­sion whether RD coordinates should be replaced by a different map projection. At the heart ofthe discussion is that many users find it difficult to work directly with geographic coordinates

74 10. Dutch national reference systems

NAP H

RD x, y

Bessel (1841)

RD map projection

NTv2 grid

ETRS89

h

Geoïd model

User environment: input & output

Processing environment

Figure 10.5: NTv2 transformation procedure used by RDNAPTRANS™2018. The figure outlines the relationshipsand transformations between ETRS89, RD2000 and NAP using the proposed NTv2 procedure, in variant 2 ofRDNAPTRANS™2018 where the datum transformation is included in the correction grid. The coordinates belowthe line are used only for computational purposes and should never be published or distributed to other users.

(latitude and longitude) and prefer working with rectangular 2D grid coordinates, but lack theexpertise and software to do the conversion to RD.

As the result of this discussion the RDNAPTRANS procedure has been modified and isbased on the Canadian NTv2 correction procedure (National Transformation version 2) that isbetter supported by existing softwares. As shown in Figure 10.5, the NTv2 procedure employsa correction grid to convert latitude and longitude in ETRS89 directly to latitude and longitudeon the Dutch Bessel 1841 ellipsoid, which are the input for the RD map projection. Theprocedure shown in Figure 10.5 includes the datum transformation into the correction grid.There is also a variant whereby the datum transformation is still implemented as a separatestep. Other technical changes to the RDNAPTRANS procedure were the introduction of aneasier to use and more standard bi–linear interpolation method and extension of the domainover which the procedure is valid. These technical changes, apart from the introduction of thenew and improved NLGEO2018 quasi–geoid and improved transformation parameters, werecarried out in such an way as to maintain consistency at the milimeter level with previousRDNAPTRANS versions. The new NLGEO2018 quasi–geoid represents a real improvement forthe height, but even so, consistency with the previous RDNAPTRANS for the heights is still atthe centimeter level.

The new version of the RDNAPTRANS procedure, called RDNAPTRANS™2018, is mucheasier to implement than the RDNAPTRANS procedure of Figure 10.4. Also, the NTv2 proce­dure is a (relatively new) standard that is now supported by many coordinate transformationand GIS software packages, including the PROJ generic coordinate transformation software.This makes it possible to fully implement RDNAPTRANS™2018 in the PROJ transformationsoftware, that is also used by many GIS softwares. Although RDNAPTRANS™2018 can beimplemented in the PROJ transformation software, using a daisy chain of smaller transforma­tion steps, each with its own +proj string (see Section 6.4), it doesn’t have an EPSG codeyet. EPSG code EPSG:28992 only implements an approximate transformation, which is goodenough for visualizations, but should never be used to facilitate the exchange of coordinates.

Latitude and longitude in the ETRS89 reference frame is the default for the exchange of

10.2. Amsterdam Ordnance Datum ­ Normaal Amsterdams Peil (NAP) 75

geo­information in Europe, but in the Netherlands, users have the choice between latitude andlongitude in the ETRS89 reference frame and RD coordinates. However, for the transformationbetween ETRS89 and RD, only software that support the official RDNAPTRANS should be used.

10.2. Amsterdam Ordnance Datum ­ Normaal Amsterdams Peil(NAP)

The Amsterdam Ordnance Datum, in Dutch Normaal Amsterdams Peil (NAP), is the officialreference system for heights in the Netherlands. It is also the datum for the European VerticalReference System (EVRS).

10.2.1. Precise first order levellingsThe history of the Dutch height datum goes back to a bolt installed in Amsterdam’s ship­building district as early as 1556. A century later, in 1682, eight stone datum points wereincorporated in the then new locks along the IJ waterway, defining a height datum that wascalled Amsterdamse (Stadts)peyl. This datum was extended during the 18th and beginning ofthe 19th Century to include the then Zuiderzee and the large rivers, and in 1818, King WilliamI decreed the use of the Amsterdams Peil (AP) as the general reference point for water levels.At that time many different height datums were in use in the Netherlands which needed to beconnected through levellings.

Figure 10.6: Team of surveyors posing for the camera during the first precise levelling, in Dutch EersteNauwkeurigheids­waterpassing, or Rijks­Hoogtemeting, probably in 1875 or 1876. The person with the whitehat is Cornelis Lely (1854­1929), who just graduated as civil engineer in Delft (1875). Later, as minister ofinfrastructure, he introduced the bill that resulted in the Zuiderzee werken, which comprised the construc­tion of a 30 km dike forming the Ijselmeer lake, and the creation of the two western polders in the for­mer Zuiderzee. Figure from H.W. Lintsen (red.), Geschiedenis van de techniek in Nederland, De wordingvan een moderne samenleving 1800­1890. Deel VI. Techniek en samenleving. Walburg Pers, Zutphen 1995,http://www.dbnl.org/tekst/lint011gesc06_01/lint011gesc06_01_0012.php).

A series of five first order levelling campaigns has been carried out to date. The 1st na­

76 10. Dutch national reference systems

tional precise levelling dates from the period 1875­1885, including 410 already existing pointsand 2100 km of continuous levelling lines. See also Figure 10.6. The datum was based onfive remaining stone datum points in the Amsterdam locks. To distinguish the newly derivedheights from previous results the name Normaal Amsterdams Peil (NAP), the Amsterdam Ord­nance Datum, was introduced. During later periods, until the 1980’s, three more precise firstorder levellings were carried out. It saw the installation of new underground reference pointsin ­ presumably ­ stable geological strata throughout the Netherland, including several posts(nulpalen) in the vicinity of ”tide” gauges (water level gauges), the introduction of hydrostaticlevelling, and new routes, e.g. over the Afsluitdijk. On the other hand, many existing pointswere lost, including all stone datum points in the Amsterdam locks. During the 3rd levellingthe level of this last stone datum point, which soon would be lost due to construction work,was transferred to a new underground reference point on the Dam Square in Amsterdam andassigned the height NAP +1.4278 m. This datum point is in a certain sense symbolic, as theheight datum is actually defined based on the underground reference points in geologicallymore stable locations.

Figure 10.7: Levelling lines of the 5th precise levelling in the Netherlands, 1996­1999 (Figure from de Bruijne etal., 2005).

In the 1990’s it became clear that motions in the Netherlands’ subterranean strata havea major influence on the NAP grid. Geophysical models indicate that Post­Glacial uplift ofScandinavia results in a slight tilting of the subterranean strata in the Netherlands, with theWest of the Netherlands sinking by approximately 3 cm per century. This was confirmed by

10.2. Amsterdam Ordnance Datum ­ Normaal Amsterdams Peil (NAP) 77

analysis of precise levelling measurements, but the uncertainty in the data was very high, anduntil then the height of the underground datum points had never been adjusted. Because ofpolicy oriented issues, related to the protection from floods, more insight was needed into theheight changes of the underground datum points. For this reason the 5th precise levelling wascarried out between 1996­1999, see Figure 10.7. This was the first time that a combination ofoptical and hydrostatic levelling, satellite positioning (GPS) and gravitational measurements,were used. It also include ice levelling measurements on the IJsselmeer and the Markermeer.The levelling measurements still constitute the basis for the primary NAP grid. The gravitymeasurements constituted the 2nd measurement epoch of the Dutch gravitational grid. Theyserved to get an independent insight into subterranean movements. The GPS measurementsserved to enhance the levelling net over greater distances, and to connect the levelled NAPheights to ETRS89. The network of the 5th precise levelling was also connected to the Germanand Belgian networks. These connections played an important role in the establishment of aEuropean Vertical Reference System (EVRS) which uses the same ”Amsterdam” datum as theNAP grid.

In 1998 a NAP monument was created at the Amsterdam Stopera. This monument, de­signed and created by Louis van Gasteren and Kees van der Veer, consists of a NAP pillarrising through the building with on top a bronze bolt a precisely the zero NAP level, two watercolumns showing the current tide levels at Ijmuiden and Vlissingen, and a third water columnshowing the water level at the time of the 1953 Zeeland flood disaster. See the front cover.

Figure 10.8: NLGEO2004 geoid for the Netherlands, with geoid height in [m] with respect to the GRS80 ellipsoidin ETRS89 (Figure from de Bruijne et al., 2005).

78 10. Dutch national reference systems

10.2.2. NAP BenchmarksThe primary NAP grid is comprised of about 300 underground points and 70 posts (nulpalen).The underground points are not accessible to the public, but provide an as stable as pos­sible basis for measurements of the secondary NAP grid. The secondary NAP grid consistsmainly of bronze bolts, with a head of between 20 ­ 25 mm in diameter, that are fitted toa building or other structure with an appropriate stability. The heights of these bronze boltshave been determined by levelling loops with an average length of 2 km with a precisionbetter than 1 mm/km. A bronze marker is installed after every kilometer. There are about35,000 of these markers (peilmerken) installed in the Netherlands. The heights of the markersare published by RWS (https://www.rijkswaterstaat.nl/zakelijk/open­data/normaal­amsterdams­peil). These markers serve as the basis for height determinationby consulting engineers, water boards, municipalities, provinces, state, and other authorities,whereby one of these markers can almost always be found within a distance of 1 km.

GPS has not replaced levelling as much as it did with triangulation. There are two reasonsfor this; (1) GPS height are not as accurate as the horizontal positions, (2) levelled (orthomet­ric) height and GPS (ellipsoidal) height are different things. See Chapter 8 for an explanation.Therefore, the dense NAP grid will not be outdated by GPS in the foreseeable future, like itdid for the RD grid, and certainly not for applications requiring millimeter accuracy.

Figure 10.9: NLGEO2018 quasi–geoid for the Netherlands on the left, with geoid height in [m] with respect tothe GRS80 ellipsoid in ETRS89, with on the right the differences with NLGEO2004. Clearly visible the much largerdomain over which the NLGEO2018 quasi–geoid is computed (Figure courtesy Cornelis Slobbe).

10.3. Geoid models – NLGEO2004 and NLGEO2018Although it is unlikely that GPS will replace levelling alltogether, GPS can be used to obtainheights with an accuracy of about 1 − 2 cm using the RDNAPTRANS procedure, as outlinedin Figure 10.4. The transformation from ellipsoidal height to NAP heights, and vice versa,requires a correction for the geoid height.

Calculation of a geoid requires gravitational measurements over ­ in principle ­ the entireEarth. The larger scales depend mainly on satellite data, but for the highest precision atregional and national scales gravity measurements in and around the area of interest areneeded.

10.4. Lowest Astronomical Tide (LAT) model – NLLAT2018 79

The first Dutch geoid, with a relative precision of 1 decimeter, became available in 1985.In order to improve this geoid in the period of 1990­1994 some 13,000 relative gravitationalmeasurements were carried out in a grid of almost 8,000 points (1 point per 5 km2) in theNetherlands. The resulting geoid, called the ‘De Min‘ geoid, became available in 1996 andhad a precision of one to a few centimeters. This was the first accurate geoid model of theNetherlands and was used by the original RDNAPTRANS procedure.

The geoid model was improved in 2004, resulting in the NLGEO2004 model, that is used byRDNAPTRANS™2004 and RDNAPTRANS™2008, see Figure 10.8. The improvements resultedfrom using additional gravitational measurements on Belgian and German territory and a setof 84 GPS / levelling points from the 5th precise levelling to define a correction surface to thegravimetric geoid. The NLGEO2004 model has a precision better than 1 cm in geoid height.The relative precision for two points close together is approximately 3.5 mm, increasing to 5mm for two points separated by a distance of 50 km to approximately 7 mm for two pointsseparated by a distance of 120 km (de Bruijne et al., 2005). Therefore, the accuracy of GPSdetermined NAP heights using RDNAPTRANS will largely depend on the precision of the GPSmeasurement.

In 2018 a new (quasi–)geoid, called NLGEO2018, was computed. Contrary to NLGEO2004,it is based on a least­squares approach using a parametrization of spherical radial basis func­tions. This approach allowed to account for systematic errors in the gravity datasets, enablesproper error propagation, and the computation of the full variance­covariance matrix of theresulting quasi­geoid model. The model itself was computed over a much larger domain thanNLGEO2004 (it now includes the Dutch Exclusive Zone (EEZ) in the North–Sea) and basedon re­processed datasets. Also new datasets have been used, including datasets in Limburg,Belgium, Germany and shipboard and airborne gravimetry data over the North Sea. More­over, along­track geometric height anomaly differences from various satellite radar altimeterswere used. Since the data area was much larger than before it became necessary to applyso called terrain corrections, which aim to remove the high­frequency signals in the data.Another improvement is that the remove­compute­restore procedure relied on a satellite­onlygeopotential model obtained from GRACE and GOCE data. Over the land area of the Nether­lands, the precision of the NLGEO2018 gravimetric quasi–geoid is 0.7 cm standard deviation.After application of the innovation function (which aims to reduce the differences between thequasi­geoid and height reference surface) the standard deviation reduces to 0.5 cm. For theNLGEO2004 gravimetric geoid, the precision was 1.3 cm. After application of the so­calledcorrection surface this number was 0.7 cm.

Figure 10.9 shows the NLGEO2018 quasi–geoid and the differences with the NLGEO2004geoid. Differences are in the range of 1–6 cm, with a systematic difference of about 3.5cm. These differences are to be expected because the innovation function and correctorsurface are based on different GNSS and levelling datasets and the permanent tide is handleddifferently. Also, when the NLGEO2018 is used in the RDNAPTRANS™2018 procedure part ofthe differences will be resolved in the transformation parameters. Therefore, differences inthe height resulting from RDNAPTRANS versions 2004 and 2018 are much smaller, with themaximum height difference of about 2.5 cm.

10.4. Lowest Astronomical Tide (LAT) model – NLLAT2018The vertical datum for nautical maps in the Netherlands, and other countries around theNorth–Sea, is Lowest Astronomical Tide (LAT) . Lowest Astronomical Tide (LAT) is the lowestpredicted tide level that can occur under any combination of astronomical conditions assum­ing average meteorological conditions, which implies that all predicted tidal heights must bepositive (although in practice lower tides may occur due to e.g. meteorological effects). Tide

80 10. Dutch national reference systems

Figure 10.10: NLLAT2018 with respect to the NLGEO2018 quasi–geoid (Figure courtesy Cornelis Slobbe).

tables, as well as charted depths and drying heights on nautical charts, are given relative toLAT. The depth of water, at a given point and at a given time, is then calculated by adding thecharted depth to the height of the tide, or by subtracting the drying height from the height ofthe tide, with all heights and depths given with respect to LAT.

The Dutch LAT model is called NLLAT2018. The LAT surface is always below the DutchNLGEO2018 quasi–geoid, but the separation between the two is not constant and dependson the location. NLLAT2018 has been computed using hydrological and meteorological mod­els, tidal water levels from 31 tide–gauge, and the NLGEO2018 quasi–geoid. The separationbetween NLLAT2018 and NLGEO2018 is shown in Figure 10.10. The LAT reference surfacein NLLAT2018 itself is given with respect to the GRS–80 ellipsoid. The accuracy of the LATreference surface is about 1 decimeter.

11Further reading

Altamimi Z., X. Collilieux, L. Métivier, ITRF2008: an improved solution of the International Ter­restrial Reference Frame, Journal of Geodesy, vol. 85, number 8, page 457­473, doi:10.1007/­s00190­011­0444­4, 2011.

A. de Bruijne, J. van Buren, A. Kösters and H. van der Marel, De geodetische referentiestelselsvan Nederland; Definitie en vastlegging van ETRS89, RD en NAP en hun onderlinge relatie,Nederlandse Commissie voor Geodesie 43, Delft, 2005, 132 pagina’s. ISBN­13: 978 90 6132291 7. ISBN­10: 90 6132 291 X.

A. de Bruijne, J. van Buren, A. Kösters and H. van der Marel, Geodetic reference framesin the Netherlands; Definition and specification of ETRS89, RD and NAP, and their mutualrelationships, Netherlands Geodetic Commission 43, Delft, 2005, 132 pages. ISBN 90 6132291 X.

IERS Conventions (2010). Gérard Petit and Brian Luzum (eds.). (IERS Technical Note ; 36)Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie, 2010. 179 pp.,ISBN 3­89888­989­6. All software and material associated with IERS Conventions (2010) canbe found at http://tai.bipm.org/iers/conv2010/conv2010.html and at http://maia.usno.navy.mil/conv2010/.

C.D. de Jong, G. Lachapelle, S. Skone, I.A. Elema, Hydrography, Series on MathematicalGeodesy and Positioning, 2nd edition, VSSD, Delft, 2010, 351 pp. ISBN 978­90­407­2359­9.(Chapter 4: Coordinate systems).

OGP, EPSG Guidance Note 7 Part 1; Using the EPSG Dataset, OGP Publication 373­7­1, August2012, http://www.epsg.org.

OGP, EPSG Guidance Note 7 Part 2; Coordinate Conversions and Transformations includingFormulas, , OGP Publication 373­7­2, June 2013, http://www.epsg.org.

PROJ contributor. PROJ coordinate transformation software library. Open Source GeospatialFoundation, 2020, https://proj.org/.

81

AQuantity, dimension, unit

The measurement of any quantity involves comparison with some (precisely) defined unitvalue of the quantity. The statement that a certain distance is 25 metres, means that it is 25times the length of the unit metre. A quantity shall be used (defined), with appropriate unit,that provides a reproducible standard.

quantity dimension unitdistance length metre [m]time duration time second [s]mass mass kilogram [kg]

Table A.1: Three fundamental quantities with the symbol for the unit indicated between square brackets.

As listed in Table A.1, the three fundamental quantities are distance, time duration andmass. The units are given in the Système Internationale (SI), the International System ofUnits. This system was first established in 1889 by the Bureau International des Poids etMesures (BIPM). The official definitions read:

• the metre is the length of the path travelled by light in vacuum during a time interval of1/299792458 of a second

• the second is the duration of 9192631770 periods of radiation corresponding to thetransition between the two hyperfine levels of the ground state of the cesium­133 atom

• the kilogram is the unit of mass; it is equal to the mass of the international prototype ofthe kilogram

The metre, in its above definition, is linked to the second, via the speed of light in vacuum 𝑐,which equals 299792458 m/s. The speed of light in vacuum is a physical constant, and itsdetermination remains a permanent challenge to physicists; the uncertainty at present is atthe 1 m/s level. Originally the meter was established by the end of the 18th century in France;it was thought to be one ten­millionth part of the meridional quadrant of the Earth (a meridianpassing through both poles).

The angle does not appear as a quantity in Table A.1. A supplementary unit (actuallythe quantity is dimension­ and unitless) is the radian [rad] for plane angles. A full circlecorresponds to 2𝜋 rad. Other units for angles are grades (gon), a circle is 400 grad (thecentesimal system), and degrees, a circle is 360 deg, or 360∘, 1 minute of arc is 1/60 of adegree, and 1 second of arc is 1/60 of a minute or 1/3600 of a degree.

83

84 A. Quantity, dimension, unit

designation symbol powerExa E 1018Peta P 1015Tera T 1012Giga G 109Mega M 106kilo k 103milli m 10−3micro 𝜇 10−6nano n 10−9pico p 10−12femto n 10−15atto a 10−18

Table A.2: Most common powers of ten and their designation, also known as SI­prefixes.

To handle a wide range of magnitudes, standard prefixes are available for the above unitsaccording to the decimal system (as for instance ‘kilo’ and ‘milli’ for meter), see Table A.2.‘ppm’ stands for parts per million and implies a 10−6 effect, and ‘ppb’ stands for parts perbillion, a 10−9 effect.

Mass is an intrinsic property of an object that measures its resistance to acceleration, i.e.it is a measure of the object’s inertia. Force is a derived quantity. It has dimension ‘massmultiplied by length divided by time­squared’; the unit is Newton [N] which equals [kg m/s2].

Colophon

This reader has been produced initially for the 3rd year BSc course CTB3310 Surveying andMapping in the Civil Engineering curiculum at the Delft University of Technology, and lateradapted to be used also for several other courses in the bachelor and master programs.

Title: Reference Systems for Surveying and Mapping

Author: Hans van der MarelContributors: Hans van der Marel, Christian Tiberius (Section 2.4 2D examples, Chapter 7gravity, Appendix A units and the exercises at the end of each chapter)Layout: Hans van der MarelE­mail: h.vandermareltudelft.nl

Published by:Delft University of TechnologyDept. Geoscience and Remote Sensing, Faculty of Civil Engineering and Geosciences, Stevin­weg 1, 2628 CN Delft, The Netherlands.http://www.grs.citg.tudelft.nl

© Hans van der Marel, Delft University of Technology.

Release history:

1.0 8 Feb 2014 Initial edition1.1 21 Feb 2014 First release for 2014 CTB3310 course1.2a 28 Aug 2014 Interim release for CIE4606 Geodesy and Remote Sensing course2.1 7 Feb 2016 Release for 2016 CTB3310 course2.1a 22 Feb 2016 Updated release for 2016 CTB3310 course2.1b 20 Apr 2016 Updated release for CTB3425, CIE4606 and CIE4614 courses.3.1 15 Feb 2020 Third main release. Changes:

­ Section 2.4 2D examples (Courtesy Christian Tiberius)­ Section 3.3 3D similarity transformation revised­ Section 4.4 Topocentric coordinates revised­ Section 5.4 Map projection examples (Initial version ChristianTiberius)­ Chapter 10 RDNAPTRANS updated­ Chapter 10 NLGEO2018 and NLLAT2018 (Courtesy Cornelis Slobbe)­ Many minor improvements by the author, and from Christian Tiberiusand Jochem Lesparre­ Licensed under CC BY­NC­SA

File name: TRS_3­1_xelatex.tex

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